Determining Cutting Force Coefficient from Instantaneous Cutting Force in Ball-End Milling

Transcription

1 th International & 6 th ll India Manufacturing Technology, Design and Research onference (IMTDR 4) December th 4 th, 4, IIT Guwahati, ssam, India Determining utting Force oefficient from Instantaneous utting Force in all-end Milling Mithilesh Kumar Dikshit *, sit aran Puri, tanu Maity, mit Jyoti anarjee 4 * Department of Mechanical Engineering, NIT Durgapur, Durgapur - 79, dixit.mithilesh@gmail.com Department of Mechanical Engineering, NIT Durgapur, Durgapur - 79, abpuri@yahoo.com dvance Design & Optimization, SIR-MERI, Durgapur - 79, maity@cmeri.res.in 4 Manufacturing Technology Group, SIR-MERI, Durgapur - 79, ajbanerjee@cmeri.res.in bstract Specific cutting force coefficients play an important role in study of cutting forces in ball end milling. Due to the complicated geometry of ball end milling cutter the effective diameter of the tool (with active cutting edges) varies along the axis of the tool during machining which leads to the discrepancies in the prediction of cutting forces. In this paper, an improved method of identification of specific cutting force coefficient is proposed for ball end milling cutter. The cutter is discretized into finite number of axial discs along the axis of the cutter. semi-mechanistic force model is adopted to relate the cutting forces in each disc and specific cutting force coefficients are calculated by inverse method and a fourth order polynomial fitting has been obtained. Several experiments were carried out at different feed and depth of cut to calibrate the proposed identification method and the same is used for predicting the cutting forces. The shearing force coefficients have larger value at lower depth of cut and they decrease with increase in depth of cut. Edge force coefficients do not vary with the depth of cut significantly. Keywords: all end milling, cutting forces, force coefficients, polynomial fitting Introduction all-end milling is a versatile milling process and one of the most commonly used milling process in automobile, aerospace and biomedical engineering sectors and especially in die and mould making industries. The quality and precision of the final components are highly influenced by the cutting forces. The precise prediction of cutting forces helps in improving the machining performances. n accurate prediction of the cutting forces depends on the modelling of cutting forces along with cutting force coefficients. utting force model is an important process model that predicts the cutting forces components in tangential, radial and axial directions. The cutting force predictions are mainly categorised into three methods: analytical, mechanistic and numerical methods. From the literature, it was found that the recent works for the force modelling in ball end milling have focused on mechanistic approach [Lee et al. (996), udak et al. (996)]. In this approach cutting forces are assumed to be proportional to the chip crosssectional area and cutting force coefficients are the proportionality constant between the cutting forces and chip cross sectional area. onsiderable amount of literatures has been found to estimate the cutting force coefficients which are the key factors for predicting the cutting forces efficiently and accurately in ball end milling. However, in most of the research works two major approaches for estimation of cutting force coefficients have been proposed: (i) orthogonal to oblique cutting transformation and (ii) estimation of cutting force coefficients through regression analysis. In orthogonal to oblique cutting transformation approach, the cutting force coefficients are the function of shear stress, shear angle and friction angle. The coefficients are determined through the orthogonal turning experiments which are then transformed into oblique cutting edge through the transformation matrix for prediction of cutting forces in helical end milling forces [rmarego and Deshpande (99), udak, E. et al. (996)]. ltintas and Lee (998) proposed a mechanics of ball end milling ug orthogonal to oblique transformation considering shearing and ploughing effect of the ball end milling cutter. It is found that a large number of experiments were required to build orthogonal database which is a time consuming process and large number of computation would be required. nother approach is known as mechanistic approach which is a direct calibration approach. In mechanistic approach, the cutting force coefficients are calculated directly from the experiments. Feng and Menq (994) developed a D cutting force model in tangential and 74-

2 Determining utting Force oefficient from Instantaneous utting Force in all-end Milling radial direction, and established an empirical relation ug the numerical polynomial fit. Lamikiz et al. (4) studied the milling of complex surfaces and proposed a semi-mechanistic model for cutting forces. The model is only suitable for a specific cutting condition only. Gradisek J et al. (4) have shown the relationship between cutting force coefficients and axial depth of cut for down and up ball end milling. Kim et al. () proposed that the cutting force coefficients in ball end milling are the functions of cutting edge position angle. Most of the researches mentioned above, focus on determining the cutting force coefficients for a particular machining condition and a certain work-tool combination. In the present study, a mechanistic force model has been developed considering both the effect of shearing mechanism due to the chip generation process on the tool s rake face and the effects of ploughing mechanisms on the flank face based on Lee and ltintas (996). The cutting force coefficients were calculated by a comparatively new method ug the instantaneous cutting force values. fourth order polynomial fitting was adopted for determining the cutting force coefficients. Geometry of ball-end milling cutter The geometry of ball end milling cutter is shown in the Fig.. The geometry of the cutter may be divided into two parts: a cylindrical part with constant helix angle ψ, and a hemispherical part of radius R equal to the radius of cylindrical part. The cutting edge lie on the hemispherical part. The ball part of the ball-end milling cutter is discretised into a finite number of discs along the axis of the cutter. The cutting edge discretization allows the simplification of the cutting edge as a sequence of the linear cutting segments. Let us consider an elemental cutting edge P of a disc. The position of elemental cutting edge P can be characterized by elevation (z) from the tip of the tool, radial distance (R z ) from the cutter axis, axial immersion angle (κ), and radial immersion angle () as shown in the Fig.. The position of the i th element on the j th cutting edge with elevation z may be given by: a b Fig.. (a) Front view of discretized cutter, (b) top view and position angles of cutting edge ( z) ( j ) ( z) = θ + φ p δ i () Where θ is the spindle rotation angle, ϕ p is the pitch angle defined by ϕ p = π/n and δ i (z) is radial lag angle varying between and δ, and varies with the local helix angle ψ i for i-th element. The value of δ is constant for a specific tool and depends on helix angle ψ and tool radius R. The local radius R (z) of each disc can be given by: ( ) ( ) R z = R R z () The axial immersion of the cutting edge P can be given by: κ = arc R z () R Where κ is the axial immersion angle of the cutting edge on the disc measured with the Z-axis from the centre of the ball part. s shown in the Fig., the position of each discrete element i of the cutting edge j can be given as the function of angular position of the elemental cutting edge and the local radius as given below. x = R zi ( ) y = Rzi cos( ) (4) Rδ z i = tan ( ψ ) In vector form the position of each cutting edge element can be given by the following formula: r ( ) = x iˆ + y ˆj + z kˆ () R R iˆ δ = δ cos ˆj i kˆ i i + + tan ( ψ ) The radius of each discrete disc can be calculated as the function of lag angle ug Eq. () and () as: δ ( ) i R i δi = R (6) tan ( ψ ) fter defining the position of each elemental cutting edge its edge length can be given by: R ds = dr = R i ( δi ) + R i ( δi ) + dδ ( 7) tan ψ ( ) It is necessary to find the uncut chip thickness at any cutter point in the engagement region to determine the differential cutting forces. Once the position of the each discrete cutting edge element is defined, it is necessary to calculate the uncut chip thickness based on the analysis of the cutter geometry modelling as shown in the Fig.. The chip thickness formulation proposed by Martelotti for the straight end mills has been modified, ce the chip thickness varies as the depth of cut and radial immersion of the cutter changes. The uncut chip thickness is expressed on the basis of radial and axial immersion angles as shown in Eq. (8). 74-

3 th International & 6 th ll India Manufacturing Technology, Design and Research onference (IMTDR 4) December th 4 th, 4, IIT Guwahati, ssam, India ( ) f t ( ) ( ) t nj j, κ = j κ. ε (8) Where f t is the feed per tooth and it is determined from feed rate and ε is the factor which shows the engagement/disengagement of the cutter with workpiece at a particular instant of rotation. a b Fig.. (a) Idealized chip thickness (b) ctual chip thickness and its projection in ball end milling Mechanistic force model In this paper cutting forces are modelled through the two fundamental factors namely shearing effect taking place in the shear zone and edge effect induced due to the ploughing at the cutting edge. Due to the complicated geometry of ball-end milling cutter, the cutting forces vary in different discs. For each axially discretized disc, the cutting force is regarded as an independent element. The total force acting on the cutter may be evaluated through the numerical integration of the forces acting on each discretized disc along the axis of the tool. s per the mechanistic method, the cutting force components may be expressed as under Lee P, ltintas Y (996): ( θ, ) = + (, θ, κ ) ( θ, ) = + (, θ, κ ) ( θ, ) = + (, θ, κ ) df tj z KtedS Ktctnj db dfrj z KredS Krctnj db df aj z KaedS Kactnj db (9) Where df tj, df rj, df aj are the tangential, radial and axial force components acting on the j-th cutting edge, K te, K re, K ae (N/mm) are the edge force coefficients or specific edge coefficients, K tc, K rc, K ac (N/mm ) are the shear specific coefficients, ds (mm) is the length of each discrete element, t nj (mm) is the uncut chip thickness (t n ) for j-th cutting edge and db (mm) is the elemental chip width in each cutting edge element. Six specific cutting force coefficients needed to be identified through the experiments for a particular cutter-workpice material under specific cutting conditions. Thus, the three elemental cutting force components df t, df r df a acting on the infinitesimal cutting edge, are determined in an orthogonal system of cutting. The actual forces in oblique cutting are determined further through coordinate transformation of the above forces. Finally, these three cutting forces F t, F r and F a (as mentioned in equation 9) are evaluated through numerical integration. 4 Specific cutting force coefficients semi-mechanistic force model is adopted and it is assumed that the average cutting force from the experiments are the input factors to derive the specific cutting force coefficients. The analytical equation for average cutting force is z F ex ( t, r, a ) = φ φ df(,, ) (, z ) d st z t r a φ φ φ () p Where ϕ st and ϕ ex are the start and exit radial immersion angles, respectively. The axial integration limits for each cutting tooth can be determined from cutter-workpiece engagement area. In a slot milling ϕ st = and ϕ ex = π. The instantaneous force at radial immersion angle can be given as, Gradisek J et al. (4), ( ) ( ) ( ) Ft f F t r = T + T F a () Where s and s represent the influence of cutter geometry on the average cutting force (due to shearing action) and average edge force (due to ploughing) respectively. They are called the geometric constants and are determined by the cutter geometry and can be given by: z z z, κ ( ), cosκ ( ) = dz = z dz = z dz z z z z z z ( ), κ ( ) ( ), cosκ ( ) ( ) ( ) = ds z = z ds z = z ds z z z z 74-

4 Determining utting Force oefficient from Instantaneous utting Force in all-end Milling [ ] ( ) ( ) ( ) ( ) ( ) ( ) K ( ) K ( ) K tc K rc K ac T = Ktc Krc Kac ac rc [ ] ( ) ( ) ( ) ( ) ( ) ( ) Kte cos Kre Kae T = Kte Kre cos Kae cos Kae Kre The average force from the instantaneous forces are given by: Ft Ktc K te f t Fr = [ T ] K rc + [ T4 ] K re () φ F p φ K p ac K a ae Where matrices T and T 4 depends on the cutter geometry, radial immersion angle and cutting depth. The matrices would be obtained by two independent integration related to start and exit radial immersion angle and axial depth of cut as given below. ( ) ( ) T = ( ) 4 T 4 = 4 4 Where c s are the immersion constants and depends on the immersion angles and can be given by: ex ; ex = ; cos ex = st 4 = st 4 st 4 ex = ; cos ex = ( 4) st st For slot milling the value for the immersion constants are: π = ; = = 4 = ; = () From the Eq. (4) we observed that the average cutting force has action called shearing action and edge action per cutter tooth is a linear function of feed and can be written as: F ( t, r, a) = F s ( t, r, a) f + F e ( t, r, a) (6) t Forces due to shearing action and edge action can be obtained experimentally from milling test at different feed rates keeping other parameters viz. axial and radial depth of cut and cutting speed. There are two unknowns and one equation as shown in Eq. (6), therefore linear interpolation method, George W. ollins () is adopted to calculate the specific cutting force coefficients at different feed rates keeping cutting speed, radial and axial depth of cut constant. Suppose the discrete cutting depths in experiments are {x, x, x, x 4, x } with feed rates of { f, f, f, f 4, f } and {F, F, F, F 4, F } is the corresponding cutting forces. While F in Eq. () at different depths should be changed into F ( t, r, a) = F ( t, r, a), N F ( t, r, a), N (7) N is the current cutting depth and N- refers to the previous cutting depth. Since the cutting force components are in three axis, each element of {F, F, F, F 4, F } has three components as well. The force on each disc is recalculated based on the forces on the current and adjacent layers. Taking one force components in tangential direction as an example, the cutting forces adopted in different discs are: F t F Ft F t F Ft F F t = (8) F F F t t N N N Where {F, F, F, F N } are input element of Eq. (8) which are considered as the cutting forces in the {,,, N} th disc. utting force coefficients are the differential value corresponding to the each disc. The coefficients are fitted with experimental data through 4 th order polynomial fitting. The detailed expression for any cutting force coefficient may be expressed as: 4 K ( t,r,a = P + P + P + P z + P ) z z z 4 (9) P, P, P, P 4, P are the fitted coefficients and z (mm) refers to the distance along the axial direction. Fig. Experimental setup Simulation and experimental results In order to determine the specific cutting force coefficients a series of slot milling test were performed on Mikron-VP 7, -axis vertical milling machining center. The cutter was two fluted solid carbide ball end mill cutter with mono layer of TilN coating from oromill Plura series of Sandvik with mm diameter, 74-4

5 th International & 6 th ll India Manufacturing Technology, Design and Research onference (IMTDR 4) December th 4 th, 4, IIT Guwahati, ssam, India 7 mm projection length and degrees helix angle. The workpiece material was aluminium block (l4- T6) of size 6 mm. -component dynamometer (Make: Kistler, model 97) was used for force data acquisition. The dynamometer has been fixed to the machining centre ug the fixtures along with the aluminium block. The cutting forces are sensed by the piezoelectric transducers within the dynamometer and an electric charge is formed due to the dynamic forces generated in the cutting edge. The charge obtained from the dynamometer is the sum of all dynamic differential forces acting on both flutes. multichannel charge amplifier of type 7 is used for converting this electric charge to voltage output. The sensitivity values for the three channels (x, y and z) in the amplifier were -7.87, -7.9 and -.69 p/n respectively. Data has been collected for full engagement region and sampling frequency rate of Hz for all tests. n inbuilt software called dymoware was used to display and record the measured data. The experimental setup is shown in Fig.. The ball end milling cutter was divided into five segments axially from.-.,.-.6,.6-,..4 and.4-.8 [mm] and a series of feed rate with.,.7,.,.7 and. [mm/tooth] were applied. Thus, six specific cutting force coefficients were calculated as discussed in the section 4. The shear and edge specific cutting force coefficients are shown in the Fig. 4. Shearing force coefficients (N/mm ) Shearing force coefficients (N/mm ) - Ktc Krc Kac Depth of cut (mm) - Fig. 4(a): Experimental data of specific shearing coefficients Ktc Krc Kac -.. Depth of cut (mm) It is found that the shearing coefficients vary largely with the cutting depths, and variations of edge coefficients with cutting depths are relatively small. s shown in the Fig. 4 (a), the specific shear cutting force coefficient fluctuates very much with depth of cuts. The value of specific cutting coefficients are very large at smaller depth of cut. lso, as is evident from Fig. 4 (c), the specific cutting edge coefficients have small fluctuation range compared to former. This phenomenon can be justified by material removal mechanism on the rake and flank contact with the work material. The verification result is shown in the Edge force coefficients (N/mm) utting depth (mm) Fig. 4 (d) Fitting of Specific edge cutting force coefficient Fig. (a-c). for the cutting condition, feed of.7 mm/tooth, depth of cut.6 mm, radial depth of cut.7, cutting speed m/min and spindle rotation rpm based on the proposed specific cutting force identification procedure. The predicted results were compared with the measured cutting forces obtained by dynamometer. The compared region is shown with the circle for one time period of the rotation of the ball end milling cutter as shown in the Fig. (a-c). It was found that the predicted results is in well agreement with the measured cutting forces. There is small deviation in the predicted cutting force components to the measured ones and is below 7% which is quite satisfactory. These deviations can be caused my machine tool vibration at higher spindle rotation and due to force measuring device during the experiments. Fig. 4(b): Fitting of specific shearing coefficients Edge force coefficients (N/mm) Kte Kre Kae Depth of cut (mm) Fig. 4(c): Experimental data of specific edge coefficients 74-

6 Determining utting Force oefficient from Instantaneous utting Force in all-end Milling Fig. comparison of experimental and predicted cutting forces for ball end milling cutter, f=.7 mm/tooth, p=.6 mm, e=.7 mm, Vc= m/min, N= rpm. (a) Tangential cutting force (Fy), (b) xial cutting force Fz and (c) Radial cutting force. 6 onclusions n improved identification procedure is proposed to determine specific cutting coefficients for ball end milling cutter. semi-mechanistic force model was adopted to derive the expression of cutting force components for axially discretized ball end milling cutter total force acting on the cutter was obtained by the numerical integration of all the elements along the tool axis. The specific cutting force coefficients were obtained by the inverse method and fitted in 4 th order polynomial and finally the specific force coefficients identification approach was validated with experimental results. The force coefficient identification process discussed in the present paper is fast and more accurate compare to conventional identification method. The effectiveness of the identification process was demonstrated through experimental and analytical derivations and forces obtained by identification process and experiments are in well agreement and it can be further use in the study of tool inclination and dynamics of ball end milling. References ltintas, Y. Lee, P. (998) Mechanics and dynamics of ball-end milling. SME Journal of Manufacturing Science and Engineering, :

7 th International & 6 th ll India Manufacturing Technology, Design and Research onference (IMTDR 4) December th 4 th, 4, IIT Guwahati, ssam, India Lamikiz,, L. N. Lo pez de Lacalle, J.. Sa nchez, M. Salgado,. (4) utting force estimation in sculptured surface milling. International Journal of Machine Tools & Manufacture, 44: 6. rmarego, E.J..Deshpande, N.P. (99) omputerized end-milling force predictions with cutting models allowing for eccentricity and cutter deflections. IRP nnals,4: 9. udak, E. ltintas, Y. rmarego, E.J.. (996) Prediction of milling force coefficients from orthogonal cutting data. Transactions of SME Journal of Engineering Industry, 8: 6 4. Feng, H.S.; Menq,.H. (994) The prediction of cutting forces in the ball-end milling process I: Model formulation and model building process. International Journal of Machine Tools & Manufacture, 4: George W. ollins, II, Fundamental numerical methods and data analysis. ase Western Reserve University, leveland, US (). Gradisek J, Kalveram M, Weinert K (4) Mechanistic identification of specific force coefficients for general end mill. International Journal of Machine Tools & Manufacture, 44: Kim GM, ho PJ, hu N () cutting force prediction of sculptured surface ball-end milling ug Z-map. International Journal of Machine Tools & Manufacture, 4: Lee P, ltintas Y, Predictions of ball-end milling forces from orthogonal cutting data. Int J Mach Tools Manuf 996; 6 (9):