FORCES, VIBRATIONS AND ROUGHNESS PREDICTION IN MILLING USING DYNAMIC SIMULATION

FORCES, VIBRATIONS AND ROUGHNESS PREDICTION IN MILLING USING DYNAMIC SIMULATION Edouard Rivière (1), Enrico Filippi (2), Pierre Dehombreux (3) (1) Faculté Polytechnique de Mons, Service de Génie Mécanique, Rue du Joncquois,53,B-7000 Mons, Belgium, tel : +32 65374547, fax : +32 65374545, edouard.riviere@fpms.ac.be (2) Faculté Polytechnique de Mons, Service de Génie Mécanique, Rue du Joncquois,53,B-7000 Mons, Belgium, tel : +32 65374546, fax : +32 65374545, enrico.filippi@fpms.ac.be (3) Faculté Polytechnique de Mons, Service de Génie Mécanique, Rue du Joncquois,53,B-7000 Mons, Belgium, tel : +32 65374548, fax : +32 65374545, pierre.dehombreux@fpms.ac.be Abstract Several methods to predict vibratory behavior in machining are listed in the literature. Some of them consist in the linearization of the machining process; they lead to the traditional form known as of the «stability lobes». This approach does not take into account some characteristics of the milling process (periodic variation of chip thickness, entries and exits of the tool), especially in finishing. Dynamic simulation of the process, based on three fundamental pillars (modeling of the cutting forces, modeling of the machined surface and prediction of the relative movements between the part and the tool), are more suitable in this case. The purpose of this article is to present a simulation tool that combines a model of cutting forces proposed in the literature by Engin and Altintas with the generation of the machined surface by a model "eraser of matter". The computer program discretizes the geometry in elementary discs along Z axis (model 2 D ½) and the dynamics is currently modeled as classical "mass-spring-damper" system. It predicts the cutting forces, the vibrations and the geometry of the machined surface. It is thus possible to give acceptable range for parameters such as spindle speed or depth of cut according to technological criteria (roughness after machining, maximum effort on the cutter, maximum vibration level). Keywords : Chatter vibration, dynamic simulation, regenerative effect, surface modeling 1 Forewords Machining is one of the most common ways to manufacture mechanical components. It allows getting high precision part of various forms and can be very flexible. High speed techniques strengthen advantages of machining and are nowadays impossible to circumvent for competitiveness reasons. Evolution of modern machine tool design tends to favor appearance of self excited vibrations [1]: - due to higher speed of the elements (e.g. slides) mass are lowered, so stiffness is also lowered; - versatile tools with longer overhang are more flexible; - higher spindle speeds tend to solicit structure in higher frequencies. 1

Machining instabilities also called 'chatter' were first studied in the fifties [2]. The most common explanation for chatter instability is based on regenerative effect. In any machining operation, the tool removes the chip from a surface that has already been machined. Chip thickness varies then due to difference of vibration between current and previous pass. For some combination of spindle speed and depth of cut, vibration level can be high. These vibrations lead to poor surface quality, increase cutting forces, accelerate tool wear and increase tool breakage risk. Two main paths were followed to control that phenomenon: online detection and prediction. Online detection techniques try to measure instability using signal given by various sensors (force sensor, accelerometer, microphone and so on). Simulation techniques simulate the machining system and try to anticipate vibratory behavior in order to compute optimal parameters for a given operation (spindle speed, depth of cut and so on). 1.1 Online detection Online detection technique philosophy is to monitor milling system using a sensor to detect the appearance of chatter and allow the operator to correct parameters to reach a stable zone. Many authors used that way to combat chatter (see for example [3],[4],[5]). Various signals are used to monitor machining behavior (spindle current, acceleration of the workpiece, cutting force, noise level). The signal is then analyzed in terms of amplitude, variance or frequency content to assess appearance of unstable conditions. Some techniques also give information of ideal parameter to be tested to reach a stable zone (see for example the magic speed proposed by Harmonizer method). 1.2 Prediction techniques Simplest prediction techniques linearize the machining process. The system is a single degree of freedom, orthogonal cut is assumed and the cutting forces are proportional to chip thickness. This approach leads to the well-known 'stability lobes' (see Figure 1) that separate spindle speed - axial depth of cut charts into stable and instable zones. Figure 1 : Typical stability lobes shape Figure 2 : Two kind of lobes given by semidiscretization method[8] Analytical methods based on this theory were developed to simulate milling [6] and improved to take more complex systems into account (for example cutter with non uniform pitch [7]). 2

These analytical method are even so limited by the nonlinearities of the milling process (periodic variation of chip thickness, entries and exits of the tool,...). These nonlinearities can have dramatic impact on stability, especially when radial depth of cut is small (e.g. in finishing). Other prediction techniques based on the study of delay differential equation [8] show the appearance of new stability lobes shape (see Figure 2) while cut is sporadic. This obviously shows that techniques taking discontinuous cut into account are more suitable to study milling process. Dynamic simulation of the milling system can take all the nonlinearities into account to get a simulation tool which is able to predict ideal cutting conditions. 2 Dynamic simulation of milling process Dynamic simulation of milling process is based on three fundamental pillars: modeling of the cutting forces, modeling of the machined surface and prediction of the relative movements between the part and the tool. These three elements are highly coupled (see Figure 3): - cutting forces are input for the dynamic model of the system to get relative movements between tool and workpiece; - displacements given by this model can be used to describe milled surface, and thus chip thickness can be computed; - chip thickness is one of the main influence factor for cutting force assessment. Efficient modeling of these three aspects can lead to an optimal global model. Figure 3 : Coupling between efforts, geometry and dynamic 3

2.1 Cutting forces prediction Many cutting forces models are listed in the literature. Simplest models consider forces to be proportional to chip thickness. Some authors model the efforts as a non-linear function of chip thickness [9]; other authors proposed models that take friction into account [10]. Sometimes, the forces are computed with curve fitting from cutting tests [11]. In our software, cutting forces are modeled following model given by Engin [12]. The tool is divided in slices of infinitesimal thickness dz (see Figure 5). Elementary efforts df t, df r and df a (tangential, radial and axial direction) are computed for each disk using six cutting coefficients: dft dfr dfa = K = K = K te re ae ds + K ds + K ds + K tc rc ac h db h db (1) h db The edge cutting coefficients K.e are constants and ds is the local cutting edge length. The second terms are associated to chip section (h is the chip thickness and db the projected length of an infinitesimal cutting flute in the direction along the cutting velocity). The K.c coefficients depend upon material; they are sometimes called specific pressure. Figure 4 : Cylindrical endmill with helical flutes Figure 5 : discretization of a tool into slices These elementary forces are then projected on a global x,y,z coordinate system and analytically integrated along the cutting tool. 2.2 Surface modeling Surface generation is based on an 'eraser of matter' model given by Peigne[13]. This model is also based on the discretization of the 3D problem in layers of infinitesimal thickness. Computation is reduced to 2D model while stiffness along cutter axis is supposed to be infinite in comparison of transversal stiffness (which is acceptable assertion practically speaking). Surface is modeled by a set of points connected by lines. The nomenclature is shown in Figure 6. 4

- point O is the center of the tool; Figure 6 : Convention (from [13]) - points A 2,A 1 and A 0 are successive positions of cutting edge at t=t -2,t -1 and t 0 ; - point B is the intersection between the OA 0 line and the workpiece surface; - point C is the intersection between the cutting tool path and the workpiece surface. The first step of computation is the evaluation of the local chip thicknes. The chip thickness is the distance between B and A 0. If point B doesn't exist, local chip thickness is null. Then, the surface is updated by erasing matter swept by the tool during interval t. 2.3 Dynamic system modeling Currently, the dynamic system modeling is very simple. The cutting tool movement (or workpiece movement) is described as a one degree of freedom system equivalent to mass-spring-damper system. Governing equations are thus: m m y x && x + c && y + c x y x& + k y& + k x y x = F x y = F y (2) Where m x,m y,c x,c y,k x and k y are modal parameters along x and y directions. The equations are solved for each step using Newmark scheme integration (h is the time step, q stand for x or y): q t+ h = q q& t t+ h + hq& = q& t t + + ( 0,5 β ) t ( 1 γ ) hq&& 2 t 2 h q&& + βh q&& t + γhq&& + h t+ h (3) Parameters β et γ are chosen so that 0,25 β 0,5 and 0,5 γ 1 to ensure unconditional stability. The simulator perform integration with β=0,25 and γ=0,5. Time step is chosen to be smaller than one tenth of the smaller natural period of the system. These parameters ensure good quality of the integration. 3 Validation testcases In order to validate the software, some simulations were performed and compared with results given in different sources from the literature. This gives indirect validation of the simulator which can then be used as prediction tool for real machining cases. 5

3.1 Testcase 1: milling effort without vibrations First validation step consist in simulation of cutting forces while system is considered as rigid. Testcase is milling of a titanium alloy with cylindrical endmill with 30 helix angle (similar to Figure 4). Data for this testcase have been given by ref [14] p 41. The endmill is discretized into 100 discs. Computing time is less than 2 seconds. Comparison between simulator (Figure 7) and figure from the reference (Figure 8) shows a good agreement. This is an obvious result because effort model is the same for both cases. Figure 7 : Forces given by the simulator Figure 8 : Cutting forces (from [14]) 3.2 Testcase 2: vibrations and surface finish prediction The second testcase simulates milling of an aluminum plate by a cylindrical mill with no helix angle. The machine and the tool are supposed to be infinitely rigid compared to the plate. The data for this example are taken from ref [13]. Dynamic of the plate is approximated as one degree of freedom system. Two sets of operating condition are examined in the article: a stable case and an unstable case. 3.2.1 Stable conditions Machining is stable for a spindle speed of 6375 RPM and axial depth of cut of 4 mm. Displacement and acceleration of the plate are shown for three rotation of the mill. Evolution is identical between simulator (Figure 9) and article (Figure 10). Figure 9 : Acceleration and displacement given by the simulator Figure 10 : Acceleration and displacement for stable condition (from [13]) 6

3.2.2 Unstable condition If spindle speed is raised to 7135 RPM, machining becomes unstable. It is possible to observe the tool jumping out of the matter. Fast Fourier transform (FFT) of the displacement shows dominant peak representing damped natural frequency (110,8 Hz), smaller peak corresponding to tooth passing period (119 Hz) and appearance of lateral peak corresponding to bifurcation (see part 4.1). Again, agreement is good between article (Figure 11) and simulator (Figure 12). Dynamic simulation is able to model surface finish. Simulation shows chatter marks on the final piece, with peaks distant from 1,5 mm. We see a good agreement between article (simulation and experimentation) and the simulator. Figure 11 : FFT of displacement given by the simulator Figure 12 : FFT of displacement, unstable condition (from [13]) Figure 13 : Machined surface given by the simulator Figure 14 : Machined surface unstable condition (from [13]) 7

4 Improvements given by dynamic simulation As seen in the foreword, classical theory for stability lobes does not take into account non linearities. As far as the cut is fairly continuous (high radial depth of cut, cutting tool with many cutting edges), classical theory is in good agreement with reality. For high speed finishing operations, the cut is more intermittent and the main assumptions of the linear theory are less valid. Dynamic simulation is now compared to other stability prediction method to show its accuracy and versatility. 4.1 Stability charts computation using dynamic simulation Stability analysis can be performed using dynamic simulation. Several couples depth of cut spindle speed are selected in the studied range. Milling system is simulated for all these different cutting conditions. The stability or instability of the cut is assessed using a threshold on a particular variable. Campomanes [15] proposes to compare chip thickness during simulation to the maximum theoretical chip thickness. If at any time step difference is greater than 25 %, instability is assessed. Figure 15 shows stability lobes for half immersion upmilling of steel computed using this method (full description of milling system is available in reference [8]). Figure 16 shows stability lobes computed using semi-discretisation of the governing differential equation. It clearly shows appearance of new kind of stability lobes next to the classical lobes. Figure 15 : Stability lobes using dynamic simulation Figure 16: Stability charts using semi-discretisation method (from [8]) Using dynamic simulation to get stability lobes allows using different stability criterion, based on technological parameters rather than on growing of perturbation. For example, it is possible to give maximum effort to limit tool wear or to avoid tool breakage. It is also possible to give a threshold for the vibration level. Roughness after machining is another criterion. Putting all these aspects on a single chart can lead to new ideal parameter selection. Using the same system as in Figure 16, and with a threshold of 150 N for the total effort, of 100µm for the displacement and of 0,0625 µm for total roughness, we get the stability charts given in Figure 17. The value chosen for the roughness is the theoretical roughness where system is entirely rigid 2 st (calculated with the classical formula Rt, th = where s t is the feed per tooth and R the cutter 8R radius).this criterion gives exactly the same result as the criterion based on the theoretical chip thickness. It is also interesting to notice that when the spindle speed is close to the natural frequency of the system, classical theory postulates that stability is maxima. However simulation shows that vibration level can be higher than in some unstable cases because the structure is excited at a natural frequency. 8

Figure 17 : Different stability criteria: total roughness (plain), effort (dash) and displacement (dot) 4.2 Frequency content of signal By studying generalized differential equation governing the milling process, Insperger[8] shows that milling system is prone to two kind of instabilities, leading to two forms of stability lobes : - classical lobes are linked to Hopf bifurcation. The movement of the system is chaotic as far as the instability is reached - the other kind of lobes is linked to flip bifurcation. The different movements can easily be differentiated using once per revolution sampling of the displacement. As an example, next two figures show displacement given by dynamic simulation of the system proposed in the article with parameters leading to Hopf and period doubling instabilities. Figure 18 : once per revolution sampling of displacement, Hopf bifurcation Figure 19 : once per revolution sampling of displacement : period doubling The authors identified theoretical frequency content of the signal using semi-discretisation method. First of all, damped natural frequency f d and tooth passing frequency f TPE (and its harmonics) which are present for both stable and unstable cutting conditions. 9

While Hopf instability is reached, new frequencies linked to the bifurcation arise. The frequency of the peaks is given by: f H 2π = ± ϖ + n n = K, 1,0,1,K(4) τ where ϖ is the dominant chatter frequency and τ the delay (time between passage of two successive teeth). For period doubling instability, the frequencies of the peaks are given by: f PD π 2π = ± + n n = K, 1,0,1,K(5) τ τ 5 Experimental validation Several machining test were performed on a Maho MH500EZ machine tool. Experiments consist in slotting and half immersion up and downmilling of steel test pieces. In order to simplify the dynamic behaviour of the system, the test pieces were clamped on a flexible structure that has been designed to be as close as possible from a single degree of freedom system (natural frequency 120,4 Hz, damping ration 0,38 %). We used a 3-fluted cylindrical carbide tool with no helix angle. The signal was recorded using accelerometer on the structure and a microphone to measure acoustic pressure. Figure 20 : Test structure holding steel parts to be machined 10

The frequency content of measured signals was analyzed to identify the dominant peaks. Figure 21 show FFT analysis of acoustic pressure recorded during machining test using parameter leading to Hopft bifurcation instability (1800 RPM ADOC 1,5 mm, half immersion downmilling). The signal is dominated by damped frequency, tooth passing period, and Hopf frequency (see part 4.2). Figure 21 : Frequency content of the signal tooth passing Frequency linked to Hopf bifurcation X damped natural frequency 6 Conclusion In this article a dynamic simulation tool for milling operations is presented. First objective of the work was to provide reliable simulations tool which is able to reproduce some testcases from the bibliography. This first step is validated using cutting forces, surface finish, stability lobes and frequency content of signal. Comparison with results from various authors shows a good agreement with the literature. Dynamic simulation thus is able to link results from different types of simulation and to reproduce typical instabilities arising in milling operation.. Dynamic simulation of the milling process can be more adapted to analyze complex milling systems. Non linearities inherent to this process (intermittent cut, tool entry and exit from the matter, nonlinear forces models, ). These simulations can lead to a better understanding of the phenomenon of dynamic instabilities. 11

7 References [1] G. PEIGNE. Etude et Simulation Des Effets Dynamiques de la Coupe sur la Stabilité et la Qualité Géométrique de la Surface Usinée : Application Au Fraisage de Profil. PhD thesis, Institut national polytechnique de Grenoble, December 2003. [2] J. Tlusty. Handbook of High Speed Machining Technology, chapter 3 : "Machine Dynamics",pages 48 153. Chapman and Hall, 1985. [3] Smith S. Delio T., Tlusty J. Use of audio signals for chatter detection and control. Journal of sound and vibration, 114:146 157, 1992. [4] Schmitz T.L. Chatter recognition by a statistical evaluation of the synchronously sampled audio signal. Journal of Sound and Vibration, 262:721 730, 2003. [5] Liang S. Y. and al. Machining process monitoring and control: The state-of-the-art. ASME Journal of Manufacturing Science and Engineering, 126:297 310, 2004. [6] Y. ALTINTAS and E. BUDAK «Analytical prediction of stability lobes in milling», Annals of the CIRP, Vol N 44, 1995, pp 357-362 [7] Y Altintas and al. Analytical stability prediction and design of variable pitch cutters.trans. ASME Journal of Manufacturing Science and Engineering, 121:173 178, 1999. [8] T. Insperger and Al. Stability of up-milling and down-milling, part 1: Alternative analytical methods. International journal of Machine tools and manufacture, 43:25 34, 2003. [9] G Stépan T. Inspeger. Stability of the milling process. Periodica Polytechnica Mechanical engineering, 44(1):47 57, 2000. [10] B. BALACHANDRAN and M.X. ZHAO A mechanics based model model for study of dynamics of milling operations Meccanica, Kluwer Academics Publishers, Vol N 35, 2000, pp 89-109 [11] W. YUN, D. CHO «Accurate 3-D cutting force prediction using cutting condition independant coefficients in end milling», Int. J. Mach. Tools Manufact., Elsevier Science Ltd, Vol N 41, 2001, pp 463-478 [12] E. ENGIN and Y. ALTINTAS generalized modelling of milling mechanics and dynamics : part I helical end mills, Acte de : 1999 ASME International Mechanical Engineering Congress and Exposition Symposium, Machining Sciences and Technology, Nashville (USA), Novembre 14-19 1999 [13] G. PEIGNE, H. PARIS and D. BRISSAUD «A model of milled surface generation for time domain simulation of high-speed cutting» Journal of Engineering manufacture Proceedings of the Institution of Mechanical Engineering Part B Vol N 217, 2003, pp 919-930 [14] Y. ALTINTAS «Manufacturing automation : Metal cutting mechanics, machine tool vibrations and CNC design, Press syndicate of the university of Cambridge, Cambridge university press, ISBN 0 521 65973 6, 2000 [15] M. L. Campomanes and Y. Altintas. An improved time domain simulation for dynamic milling at small radial immersion. Transaction of the ASME, 125 :416 422, Augustus 2003. 12