Cutting forces parameters evaluation in milling using genetic algorithm

Transcription

1 Cutting forces parameters evaluation in milling using genetic algorithm E. Rivière-Lorphèvre *, J. de Arizon, E. Filippi, P. Dehombreux Faculté Polytechnique de Mons, Service de Génie Mécanique, rue du Joncquois 53, B-7 Mons (Belgium) * edouard.riviere@fpms.ac.be Abstract Simulation of the milling process is a widespread method to improve productivity in the machining process. Several phenomena can be studied and controlled by this mean. All the simulation methods need parameters that characterize the interaction between the tool and the workpiece in order to evaluate the cutting forces. Many models were developed to link efforts to macroscopic parameters (depth of cut, feed rate), but the coefficients of these models are often difficult to find out from intrinsic properties of the materials (Young's modulus, yield strength, hardness,...). Optimisation algorithms are thus necessary to retrieve those coefficients from cutting force measurement. For linear relationships the method can be fairly simple but the non linear models require more complex optimization algorithms. The aim of this article is to set out different methods to retrieve cutting parameters for several cutting forces models. Linear models are studied with simple least square fitting method. Genetic algorithms are tested on nonlinear cutting forces models. The optimization methods are validated using both simulated and measured cutting forces in order to demonstrate its use in practice. The agreement between simulation and measure is good so the methods can be used to give input parameters for the simulation of the whole machining process. INTRODUCTION. Machining process optimisation The production of mechanical part in tight tolerance and with optimal productivity is one of the main challenges of modern production technique. Machining is a very common operation to obtain finished part of complex shapes with high precision. In a more and more competitive market, the tuning of the optimal solution must be as effective as possible. The optimisation by means of simulation is an important improvement. It allows getting the optimal cutting conditions for a given operation without interrupting the production process to perform experimental tests. Many aspects of the machining process can be studied by simulation. One of the most developed domains is the prediction of optimal cutting conditions to prevent appearance of chatter vibrations ([],[2],[3]). The simulation of the whole machining process to study the vibrations, the cutting forces and the surface finish after machining is also possible [4]. One of the most difficult parts of the simulation is the model that describes the cutting forces. Model based on intrinsic properties of the materials are difficult to establish [5], and many external influences factors must be taken into account (coating of the tools, lubrication ). The aim of this article is to set out an algorithm to retrieve parameters for cutting forces model from a single milling test. An inverse method based on least square fitting is developed for linear model. If the cutting force model is more complicated, optimization based on genetic algorithm is tested to retrieve the parameters that have a nonlinear influence..2 Cutting force models A common approach is to divide the complex shape of the cutting tool in a set of slices along its axis. Figure : discretisation of a tool into slices

2 For each slices, a cutting force model based on macroscopic parameters (feed per tooth, depth of cut ) is assumed. The elementary efforts for each slice are projected in a global reference frame and all the contribution are summed along the cutter axis to get the total resultant efforts. The coefficient describing the cutting forces models must be established from experimental measurement. Many mechanistic force models are developed in the literature ([6], [7], [8]). We chose to test two of the most common of them. The first one assesses that the cutting forces are produced by two causes: the shearing of the chip (force proportional to the undeformed chip section h.db) and the friction along the cutting edge (force proportional to the length of the cutting edge ds). The efforts along direction t, r and a (respectively tangential, radial and axial with respect to the cutter) are thus computed as () dft = Ktc h db + Kte ds dfr = K rc h db + K re ds dfa = K ac h db + Kae ds () Coefficients K.c and K.e are assumed to be constant for cutting parameters varying around nominal values. Another common model postulates that the effort is a nonlinear function of the undeformed chip thickness. The efforts in the three directions can be described by the relationship (2) Ft = K Fr = K F = a K t r a db h n db h db h 2 INVERSE METHOD n n (2) 2. Summary of the method The identification method is based on the inversion of matrix relationship between efforts (experimentally measured) and unknown cutting parameters. The model described by system of equation () can be arranged in a matrix relationship { df} = [ A] { K} (df is the vector containing elementary effort df t, df r and df a ). The matrix [A] contains data linked to the geometry of the cutter and the technological parameters. Vector {K} contains the unknown cutting forces parameters. [ A] h db = h db ds h db ds ds (3) Ktc K rc K ac K = (4) Kte K re K ae { } These local relationships can be projected on a global frame (x,y,z) (see Figure 2): cosφ sinφ sin κ sinφ cosκ B = sinφ cosφ sin κ cosφ cosκ (5) cosκ sin κ [ ] Figure 2: direction of elementary efforts and definition of angles All the relationships are then analytically integrated for all cutting edges along the axis of the cutter (nd is the number of discs along the axis of the cutter; nt is the number of tooth): [ ] = nd nt (6) C [ B][ A] i= j= The matrix [C] links for each time step the efforts to the cutting coefficients, as the system contains three equations and six unknowns, it is underdetermined. Araujo [9] proposes to inverse the relationship for two consecutive time steps in order to get an invertible system. Unortunately, this relationship is ill conditioned for some time steps, so the precision is not always adequate. A more suitable method is to assemble all the relationship for different time steps and find the best fit of the parameters by a least square fit. All the matrix relationships between measured

3 efforts and parameters of the cutting force model are assembled in a linear system. F x Fy F [ C] z 2 F [ C] = { K} x 2 (7) 2 F y 23 M 2 F [ D] z 23 M { F} i F x is the measured value of effort along direction x during time step i, [D] is the assembled matrix. The system (7) contains 3n equations (n is the number of time steps considered for the optimisation) and only six unknown. The best value of the unknown to minimize the error is obtained by the classical relationship: T T K = D D D F (8) ( ) ([ ] { }) { } [ ] [ ] 2.2 Application on a simulated testcase In order to check the adequacy of the method, we have simulated the cutting forces for slot milling with a ball-end mill (diameter 2 mm, 2 cutting edges). The feed rate is,8 mm/tooth, the axial depth of cut is 2mm (data are extracted from []). The cutting coefficients are given in Table. Simulation of the cutting forces is achieved using relationship shown in paragraph 2.. While computing cutting forces, cutting coefficients are input data and cutting forces are unknown. The simulated signal is disturbed by random white noise (the amplitude is 5% of the maximum value of the effort) and used as an input for the identification method. Coefficient Input value Fitted value K tc 272 MPa 266 MPa K rc 848 MPa 837 MPa K ac -725 MPa -73 MPa K te 7 N/mm 8 N/mm K re 8 N/mm 9 N/mm K ae -7 N/mm -8 N/mm Table : Cutting coefficients for the first testcase Table compares the coefficients retrieved by our method to the input data. The differences are small enough to assess the adequacy of our method. Figure 3 shows the evolution of the cutting forces during one revolution of the tool. Figure 3: Disturbed signal and signal computed with coefficients extracted with the inverse method 2.3 Application to measured efforts The identification method is then tested on cutting forces measured with Kistler 923B rotating dynamometer. The dynamometer is linked to the spindle and acts as a toolholder. The cutting forces are measured in three directions: along the axis of the cutter and in two orthogonal directions linked to the tool (the reference frame rotates according to the spindle speed). Figure 4 : Reference frame for cutting force measurement with a rotating dynamometer The experiments have been carried out using a high speed steel cutter (diameter 8 mm, 2 cutting edges, helix angle of 3 ) and St 52-3 steel part. The method has been tested on a series of measurements where technological parameters vary around a nominal point (spindle speed 875 RPM, feed.4 mm/tooth). A practical difficulty arises while using measured signals: the initial angular shift of the first tooth must be determined for a good adjustment, but this value is difficult to measure in practice. In order to avoid delicate empirical adjustment, we obtained this value by minimizing the RMS value of the difference between measured and computed efforts defined by relationship (9)

4 RMSerror = npoint s i= θ i i [( F F ) θ ] end c θ m begin 2 (9) The identification method is thus applied with different angular shifts. The ideal angular shift is the one that gives the smallest value of the error. We show here the results for a slotting test at mm axial depth of cut. The method was used on the measured signal and on the signal filtered by a Bessel fourth order low pass filter (cutting frequency of 3 Hz). Figure 5 shows the evolution of the RMS error with respect to the angular shift of the first tooth. The minimum of the function is easy to identify. Figure 5: Evolution of Mean RMS error with angular shift Figure 6 shows the comparison between the measured force and the result of the simulation using coefficients obtained by the identification method; Figure 7 summarizes the results obtained with the filtered signal. We can see the quality of the adjustment given by our method. Figure 6: Superimposition of measured and fitted signals for both direction perpendiculars to the axis of the cutter Figure 7: Superimposition of filtered and fitted signals for both direction perpendiculars to the axis of the cutter Coefficient Measured signal Filtered signal Ktc 3599 MPa 3476 MPa Krc 2574 MPa 257 MPa Kte 68 N/mm 7 N/mm Kre -3 N/mm -3 N/mm Table 2 : Cutting coefficient identified on both signals Table 2 gives the cutting coefficients identified on both measured and filtered signal. We can see that the coefficients are very close so our method is able to give good results, even on a disturbed signal. We also notice that if this method is applied to a series of measures with parameters varying around a nominal point, the cutting coefficient remain fairly constant []. 3 GENETIC OPTIMISATION 3. General principle Genetic algorithm can be used as an optimization method for technological parameters of production processes ([2], [3]). These problems are often highly nonlinear. The basic concept of the method is to consider each particular solution included in the parameter space as a member of a population described by a genetic code. The genes describing the individual is a binary code representing the value of each parameter. The value of the objective function is computed for each individual. If the problem is a minimizing problem, the better adapted individuals are those with the smallest value of the objective function. As in the real life, the population evolves during the time by mean of breeding or mutation.

5 The general algorithm is as follow: An initial population is randomly generated, the value of the objective function is computed for all these individuals Part of the population is selected for breeding; better adapted individuals have higher probability of selection; Selected individuals mix there genes to get the new population (children); Some of the genes are subject to mutation (one gene has its value randomly changed), this process decreases the risk of being stuck at a local minimum. The best individual of the population gives the optimal solution of the problem. The computation is stopped after a given number of generations or when the optimal solution does not change during a given number of generations 3.2 Adaptation to cutting forces parameters identification We used the genetic algorithms to identify the parameters for the cutting force model described by relationship (2). In order to limit the number of variables in the optimization process, we divided the parameters to identify in two categories: the parameters that have nonlinear influence (angular shift and exponent n); the parameters that have linear influence on the cutting forces (the three cutting coefficients K.). The genetic algorithm optimization is performed to find out the nonlinear parameters. The genetic code of the population thus describes the optimal angular shift and the exponent of the cutting force model. The parameters that have linear influence are obtained by the inverse method described on part 2. We coupled a genetic optimisation toolbox for Matlab [4] with our adjustment algorithm developed in C language. The optimisation toolbox gives values of the initial shift and the exponent of the cutting model. As the Matlab toolbox is suitable for maximisation of a function, the fitness function must be higher for low RMS error. A common approach is to define fitness function as below: Fit ( n, shift) = REF RMS( n, shift) () The reference value must be high enough to make sure the fitness value is always positive. The identification algorithm gives for all couple the value of the RMS error and the optimal cutting coefficients. Figure 8 : Exchange of information between the models 3.3 Application to nonlinear identification We tested the nonlinear fitting on the same measured signals as in paragraph 2.3. The population is composed of twenty individuals. Each variable (exponent or angular shift) are coded as a 2 bit digit. The generation gap (part of the population which is replaced by new individuals at the next generation) is fixed at 9%. A mutation probability of 2% is selected. The convergence of the algorithm can be described by the evolution of the RMS error of the best individual for all generations. Figure 9 shows the evolution of the RMS value of the best individual during time. After 5 generations, the optimal value is constant. Figure 9 : Evolution of the RMS error of the best individual for consecutive generations

6 The comparison between measured and fitted signals is given in Figure for measured signals and in Figure for filtered signals. Figure : Superimposition of filtered and fitted signasl for both direction perpendiculars to the axis of the cutter with parameters identified using genetic algorithm Figure : Superimposition of filtered and fitted signals for both direction perpendiculars to the axis of the cutter with parameters identified using genetic algorithm Coefficient Measured signal Filtered signal Kt 42,7 N/mm 2-n 34,5 N/mm 2-n Kr 26,23 N/mm 2-n 2,49 N/mm 2-n n,52,49 Table 3 : Parameters identified with the method based on genetic algorithm optimisation This method gives a mean RMS error of 35 N for the measured signals and of 26 N for the filtered signals. The identified ideal shift (24 ) is in adequacy with the value obtained by the other identification method (see paragraph 2.3). 4 COMPARISON OF THE METHODS The inverse method developed in part 2 and the method based on genetic algorithm optimisation (part 3) can both identify the parameters of a mechanistic cutting force model. Table 4 shows the comparison of the methods in term of precision. We can see that the nonlinear model gives a better adjustment than the linear model. RMS error for RMS error for linear model nonlinear model Original 47,23 N 35,3 N signal Filtered signal 3,86 N 26,3 N Table 4 : comparison of the RMS error given by the identification method The first method is very simple so the computation time is very small. The inverse algorithm must be repeated few times to identify the initial shift of the first tooth as in Figure 5. The method based on the optimisation of parameters with a genetic algorithm is more time consuming. For example, the testcase of part 3 needs 5 generation of 2 individuals to reach to the optimal solution, the inverse algorithm must be called about 9 times (the generation gap is of 9%). However, this method is much more general and allows the identification of nonlinear constitutive laws. 5 CONCLUSION In this article we describe a method to retrieve the parameters of a cutting force model from the measurement of the effort during machining. If the parameters have a linear influence on the model, the relationship between measured forces and unknown parameters can be summarized by a matrix relationship. Identification is performed by means of least square fit. For more complicated relationships, the optimal parameters that have nonlinear impact can be obtained by mean of genetic algorithm optimization. The advantage of the linear optimisation technique is that it gives simple and fast method to obtain adjustment from a single measurement of cutting forces. Although, the cutting force model must contain linear relationship between the coefficients and the cutting forces.

7 The optimisation by means of genetic algorithm needs some more computation time but is virtually able to identify parameters for any cutting force model. Both methods have been tested on measured signals, they able to give good quality adjustments even if the measure is noisy. The identified parameters can be used further to simulate the whole cutting process. 6 REFERENCES [] Tlusty, J. and Polacek, M., 963 The stability of the machine tool against selfexcited vibration in machining, ASME International research in production engineering, [2] Altintas, Y. and Budak, E., 995, Analytical prediction of stability lobes in milling. Annals of the CIRP, 44: [3] Inspeger, T. and Stépan, G., 2, Stability of the milling process, Periodica Polytechnica Mechanical engineering, 44() : [4] Rivière-Lorphèvre, E., Filippi, E. and Dehombreux, P., 27, Chatter prediction using dynamic simulation, International review of Mechanical Engineering (): [5] Moufki, A., Devillez, A., Dudzinski, A. and Molinari, A., 24, Thermomecanical modelling of oblique cutting and experimental validation, International Journal of Machine Tool and Manufacture, 44 : [6] Balachandran, B., 2, Nonlinear dynamics of milling processes, Philosophic Transaction Royal Society London Academy, 359 : [7] Faasen, R.P.H., Van de Wouw, N., Oosterling, J.A.J. and Nijmeijer, H., 23, Prediction of regenerative chatter by modelling and analysis of high-speed milling. International Journal of Machine Tools and Manufacture, 43 : [8] Engin, S. and Altintas, Y., 2, Mechanics and dynamics of general milling cutters. part I : Helical end mills. International journal of machine tool and manufacture, 4: [9] Araujo A.C. and Silveira J.C., 2, Analysis of the specifc force on end milling, proceedings of the 22nd Iberian Latin-American Congress on Computational Methods in Engineering. [] Engin S. and Altintas Y., 2, Mechanics and dynamics of general milling cutters. part I : Helical end mills, International journal of machine tool and manufacture, 4 : [] Rivière-Lorphèvre, E., Etude et simulation de procédés de fraisage grande vitesse : Efforts de coupe, Stabilité, Etats de surface, PhD Faculté Polytechnique de Mons, 27. [2] Jain, N.K, Jain, V.K. and Deb, K., 27, Optimization of process parameters of mechanical type advanced machining processes using genetic algorithms, International Journal of Machine tools and Manufacture, 47:9-99 [3] Vijian, P. and Arunachalam V.P., 27, Modelling and multi objective optimization of LM24 aluminium alloy squeeze cast process parameters using genetic algorithm, Journal of Materials Processing Technology 86: [4] Chipperfield, A., Fleming, P., Pohlheim, H. and Fonseca, C, 994, Evolutionary Computation Research Group: Genetic Algorithm Toolbox, g/gat.html