Mechanics and Dynamics of Ball End Milling

Mechanics and Dynamics of Ball End Milling Y. Altinta§ P. Lee Department of Mechanical Engineering, The University of British Columbia, Vancouver, BC, V6T1Z4 Canada

Mechanics and dynamics of cutting with helical ball end mills are presented. The helical ball end mill attached to the spindle is modelled by orthogonal structural modes in the feed and normal directions at the tool tip. For a given cutter geometry, the cutting coefficients are transformed from an orthogonal cutting data base using an oblique cutting model. The three dimensional swept surface by the cutter is digitized using the true trochoidal kinematics of ball end milling process in time domain. The dynamically regenerated chip thickness, which consists of rigid body motion of the tooth and structural displacements, is evaluated at discrete time intervals by comparing the present and previous tooth marks left on the finish surface. The process is simulated in time domain by considering the instantaneous regenerative chip load, local cutting force coefficients, structural transfer functions and the geometry of ball end milling process. The proposed model predicts cutting forces, surface finish and chatter stability lobes, and is verified experimentally under both static and dynamic cutting conditions.

true kinematics of milling (Martelotti, 1941, 1945) are considered. The dynamic chip regeneration model is similar to the Ball end milling has been used extensively in the manufactur- improved kinematics model for cylindrical cutters presented by ing of sculptured surfaces such as those encountered on dies Montgomery and Altintas (1991). The cutting forces are apand molds, turbines, propellers, and aircraft components. Prior plied to the structural dynamic model of the cutter, and the knowledge of cutting forces, surface form errors and chatter resulting static and vibration displacements are predicted using vibrations can assist process planners in selecting appropriate a numerical integration method. Later, the exact position of the cutting conditions for higher productivity and dimensional accu- tooth is evaluated by considering the true rigid body motion racy (Wecketal., 1994). and structural displacements of the tooth. The chip thickness is Mechanics of ball end milling operations have been studied calculated as a radial difference between the digitized surfaces before (Lim et a l , 1993; Tai et al., 1993; Yucesan et al, 1996). at the present and previous tooth periods. The model predicts However, those studies are limited to prediction of static cutting the cutting forces, chatter vibrations and dimensional surface forces and form errors using mechanistically evaluated cutting finish using a common cutting model and geometry. The model coefficients. Although Yucesan and Altintas (1996) used a is demonstrated in ball milling of titanium alloy Ti6AlV4. semi-mechanistic model by predicting the friction and normal Henceforth, the paper is organized as follows: Brief ball end loading of the tool at the rake and flank faces, the mathematical mill geometry, modeling of chip thickness, surface and cutting model of the cutting process was quite cumbersome and not force expressions are given in section 2, followed by time douniversal. Yang and Park (1991) and Sim and Yang (1993) main dynamic cutting system model in section 3. Stability lobes, made an assumption that each oblique segment along the ball time domain simulation and experimental results with and withend mill axis obeys the orthogonal cutting laws. Budak et al. out chatter vibrations are presented in section 4. The paper is (1996) however presented an experimentally verified model concluded with a summary of contributions by the integrated which allows transformation of orthogonal cutting parameters static and dynamic model of the ball end milling process. to oblique milling edge segments. The same model is extended to helical, ball end mill flutes in this paper. The validity of the extended oblique cutting model is proven in chatter vibration 2 Modeling of Ball End Milling Process free, static ball end milling tests in the paper. Ball End Mill Geometry. The details of the ball end mill The dynamics of general milling, the chatter vibrations, have been investigated by numerous researchers in the past. Tlusty geometry can be found in Yucesan and Altintas (1996), but its (1986) had contributed to the understanding of chatter vibra- brief mathematical model related to this paper is summarized tions in general. He and his co-workers introduced time domain in Fig. 1. The cutting edges have a helix angle of i0 at the ballsimulation models to investigate the mechanism of chip load shank meeting boundary (Fig. l a ) . Due to the reduction of regeneration and tool jumping out of cut (Tlusty and Ismail, radius in (x - y) planes towards the ball tip in axial (z) direc1981), the effect of high cutting speeds on the chatter stability tion, the local helix angle i(tj/) along the cutting flute varies (Tlusty, 1986), and efficient simulation of stability conditions for constant helix-lead cutters. The coordinates of a point on in milling (Smith and Tlusty, 1993). However, the chatter vi- the helical ball end mill geometry is defined by a vector drawn brations in ball end milling, which has complex geometry and from the tip (O) of the cutter. The local radius of the cutter dynamic chip load generation mechanism, have not received a (/?(i/0), the position of the edge (j) point, and its lag (i//), immersion ('P(z)) and actual helix (i(ip)) are defined by set similar attention in the past. of geometric relationships summarized in the Appendix. Hence, A combined model of static and dynamic ball end milling a point on the flute j at height z is tracked by its instantaneous process is presented in this paper. Since the chip load is ex- angular immersion \1/ in the global coordinate system, tremely small in most ball end milling of hard materials, the 1

Introduction

Contributed by the Manufacturing Engineering Division for publication in the

*>(z) = 0 + O " - l ) < f c , - - f l a n ;„

(1)

JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received

Dec. 1995; revised Aug. 1997. Associate Technical Editor: C. H. Meng.

where 0 is the rotation of the reference flute j = 1. 9 is measured

684 / Vol. 120, NOVEMBER 1998

Transactions of the ASME Copyright © 1998 by ASME

Downloaded From: http://manufacturingscience.asmedigitalcollection.asme.org/ on 12/19/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use

\Y Relief angle

a,

\ i

\.

•V-* AX

\ \

Rake C(z) angle

! i

'; dFt 1 * X

Z CM

^ _

1

dFa

A

dFr

Nf-3

Nf-2 Fig. 1 Geometric model of the ball end mill

clockwise from +y axis, and from the center point O. In the simulation model, the cutter is divided into Kz discrete axial elements, each having a thickness of Az = alKz where a is the

axial depth of cut. The corresponding angular immersion for each cutting edge point in contact with the material is identified, and is applied in the instantaneous chip thickness calculation.

Nomenclature a = axial depth of cut db = differential cutting edge length dz = differential height in axial direction dF,j, dFrJ, dFaj = differential cutting forces in tangential, radial, and axial directions for tooth j f = feed speed [mm/sec] h = uncut chip thickness normal to cutting edge in milling z'o = helix angle at ball shaped flute and shank meeting point i'(t/0 = local helix angle or angle of obliquity on the flute x, y, z = coordinates of a point on the cutting edge

xs, ys, zs = coordinates of a point on the cut surface x'(t), y'(t) = displacements of the cutter center due to structural vibrations at time t Fxj, Fyj, Fzj = milling forces in Cartesian coordinates on flute j Hx, Hy [m/N] = Direct transfer functions in x and y directions. K,c, K,r, KlK. = cutting force coefficients [force/area] Ku, Kle, Kae = edge force coefficients [force/length] Kz = number of axial elements N, = number of flutes '/ N = spindle speed

Journal of Manufacturing Science and Engineering

R0 = ball radius R(tp) = tool radius in x-y plane at a point defined by ip R„, Rc = Radial distance between the stationary spindle center and points on the cut surface and cutting edge, respectively 4>(z) = lag angle between the tip (z = 0) and a point on the helical flute at height z * = immersion angle in global coordinates, measured from y-axis (CW) tpo = maximum lag angle between the tip (z = 0) and uppermost cutting point (z = a) 6 = tool rotation angle, measured from y-axis (CW) 4>p = pitch angle of the cutter (=2ir/ Nf)

NOVEMBER 1998, Vol. 120 / 685


SURF(m,x,y,z)

F^CMO-R^nyii+l)

] SURF(m,x) Fig, 2 Evaluation of chip thickness from the generated surfaces at present and previous tooth periods

Chip thickness Model. The axial depth of cut and chip load are usually very small in ball end milling operations, especially in machining hard die and tool steels. In order to evaluate the cutting forces accurately, the true kinematics of milling (Martellotti, 1941) are considered. The mathematical model resembles the earlier dynamic face milling work presented by Montgomery and Altintas (1991), and only the modifications required for ball end milling are given here. The cutter is divided into slices in the axial direction, and the surface cut by the helical ball end mill slice at each axial level is digitized by a number of points, and stored in Cartesian coordinates, see Fig. 2. The center of the coordinate system is selected as the stationary spindle center (0, 0, 0), and the table moves in negative x direction with a feed speed of / [mm/s]. The coordinates of the cutter center are determined by the vibrations in the feed and normal directions: xc(t) = x'{t)

ydt) = y'(t)

(2)

where [x'(t), y'(t)] are the static or dynamic displacements of the cutter center due to structural flexibilities at time t. The angular coordinate 8 is 8(t) = UJ X t, where the angular speed is UJ [rad/s] = 2TTN/60, and N [rev/min] is the spindle speed. In order to model the rotational and structural displacements of the cutting edges in discrete time domain, the process is digitized at At [s] time intervals. The corresponding discrete angular rotation interval becomes A8 = ujAt. In the beginning of the simulation, the cutter is assumed to reach its steady state immersion ( 0 , ) without any vibrations. The arc of cut is divided into M = &JA9 angular segments, and the corresponding points on the arc of cut are stored in an array for each discrete axial element, see Fig. 2. The digitized surface is represented by M points at each axial layer, each having the coordinates SURF(P(m), ik), where point P(m) = [x(m), y(m)} is located at axial depth it = kzAz • For example, the in plane coordi686 / Vol. 120, NOVEMBER 1998

nates of the digitized surface point at an axial depth of cut z = ktAz is stored in the computer as: xs(m) = SURF(m, Zk, x)

(3)

ys(m) = SURF(m, Zk, y)

The surface at axial layer Zk is represented by points m = 0, 1, 2, . . . M, and their x coordinates are updated at each time interval Af by an amount of the incremental feed motion / X At. The integrated static-dynamic milling system is simulated at discrete time intervals At. During the first rotation of the cutter, static and dynamic displacements are not considered, therefore a smooth surface is generated by the rigid body motion of the milling system. In the subsequent revolutions, the structural displacements are considered. The cutting edge penetrates into the workpiece due to the rigid body feed motion combined with the structural vibrations of the milling system. At time t, the coordinates of a cutting edge point are calculated for each axial depth of cut Zk as, x{kz, t) = R{kz) sin ( * ) + y(kz,t)

x'{t)

= R(kz) cos (•$) +y'(t)

•

(4)

z(kz, t) = kzAz The radial distance between the center of the spindle and the cutting edge point is, Rc(kz,

t) = Jx(kz,

t)2 + y(kz,

t)2.

(5)

Depending on the magnitude and velocity of the vibration, the cutting edge point may be anywhere in the x - y plane of the cut. The two points, which are generated in the previous tooth period and are closest to the new cutting edge location, Transactions of the ASME


Surface created by tooth (j-1 Fig. 3

Dynamic model and chip thickness regeneration mechanism in milling

are identified by searching the previous surface array, see Fig. 3. Using a linear interpolation between the two points, the point P„{m) = {xs(m),y,(m)}, which lies on the radial vectorR c {k z , t) but on the previously generated surface, is evaluated. The radial distance between the point Ps(m) and the spindle center is given by,

Modeling of Cutting Forces. A set of curvilinear coordinate system normal to the ball envelope is used to specify the resultant cutting forces acting on the flute. The elemental tangential, radial, and axial cutting forces dF,, dFr, dFa acting on the cutter are given by

RAkz, m, t) = {xXkz, m, t)2 + yAK-. m, t)2

dF,iO, z) = Ku,dS + K,M6, V> «)db

(6)

The actual chip thickness is found by projecting the difference in the radial distances on the line which is passing through the ball center (Figs. 1 and 2), hit) = [Rc{kz, t) - R,(kz, m, t)] sin K

(7)

where K = sin""1 (R(I/J)/R0). As the cutter rotates at discrete (A9) intervals, the material, which is swept by the cutting edge due to its rigid body motion and structural displacements, is identified and the surface points are updated. Thus, at any time, the surface array SURF has M number of points per axial level which represent the instantaneous arc of cut. Since the axial depth of cut is divided into Kz number of layers, M X Kz number of points represent the entire, spherical shape cut surface. The model has several advantages: • •

•

•

When the vibrations are neglected, the static milling can be simulated. The finish surfaces are predicted by keeping the points which have zero and full immersions, $ = 0 and \P = %. The generated finish surface represents feed marks, static form errors or chatter vibration errors depending on whether rigid, statically or dynamically flexible milling system structure is assumed, respectively. The resulting chip thickness h(f) represents the true regenerated value. The influence of structural static and dynamic deflections of the tool on the immersion and chip load is automatically considered by the model. The nonlinearity in the chatter, i.e. the tool jumping out of cut, is accounted for by imposing the condition: if h(t) < 0 then h{t) = 0.

Journal of Manufacturing Science and Engineering

dFrie, z) = K,.JS + Krchid, ijt, K)db dFaiO, z) = KmdS + Kachi6, iji, K)db

(8)

where hid, ip, K) is the uncut chip thickness normal to the cutting edge, and varies with the position of the cutting point according to regenerated surface, see Eq. (7) and Figs. 1 and 2. The cutting forces are separated as edge (e) and cutting (c) components. The edge force coefficients (AT,e, K„, Kae) are in [N/mm], constant and lumped at the edge of the flutes. dS is the differential length of the curved cutting edge segment given by Eq. (A.7) in the Appendix. The differential width db is the projected length of an infinitesimal cutting flute in the direction along the cutting velocity. The relationship between db and dz is given as db = dz/s'm K. The cutting coefficients KK, Krc, Kac are identified from a set of orthogonal cutting tests using an oblique transformation method. The shear stress, shear angle and friction coefficient of the workpiece material cut by a tool material are measured from orthogonal cutting tests. By assuming that chip flow angle is equal to the local helix angle at the analyzed edge segment, an oblique cutting mechanics transformation is applied on orthogonal cutting parameters. The details of the transformations are given by Budak, Altintas and Armarego (1996), and summarized in Appendix B. The differential cutting forces acting on the cutting edge are calculated for each local chip load according to Eq. (8), and transformed to the Cartesian coordinates via the transformation matrix T, NOVEMBER 1998, Vol. 120 / 687


{dF)

[T]{dF} r t a

spindle speed), machine tool dynamics, cutting coefficients, and other miscellaneous variables such as number of digitized points on the cut surface and simulation time step. The simulation time step (sampling frequency) is selected to capture the highest vibration frequency in interest. Each simulation runs in a series of small time steps for the chosen duration. At each instant, the cutting forces and tool deflections are recomputed, the surface geometry is updated and stored. The simulation program predicts milling forces, dynamic displacements of the machine tool structure at the tool tip, and the finished dimensional surface profile.

dFx dFy dF7 -sin -sin

(K) (K)

sin (\P) cos ( * )

cos

(K)

-cos (\P) sin ( * ) 0

-cos (K) sin (\I/) -cos ( K ) cos (\E>) -sin («•) dFr dF, dFa

(9)

The instantaneous cutting forces acting on one flute are found by digitally summing the differential forces contributed by all elements along the axial depth of cut (a), i.e. kz = 1 . . . Kz. The total forces acting on the cutter body are evaluated by summing the contributions of all flutes which are in the cutting zone. Note that the cutting force coefficients (K,c, Kac, Krc) may be dependent on the local chip thickness, which is considered in the orthogonal cutting parameters, see Appendix B. 3

Structural Dynamic Model Since the depth of cut is small in ball end milling operations, the transfer function of the ball end mill attached to the spindle is measured at the tool tip in both feed (JC) and normal (y) directions, see Fig. 3. The cutter-spindle assembly is assumed to be rigid in the axial direction. There may be several spindle and cutter modes in both directions, which are measured and identified using experimental modal analysis techniques. The measured transfer function (//,) in feed (x) direction can be expressed as: HAs) =

x'(.s) Fx(s)

a-xi + PxiS

(10)

+ 2^;UV,,5 + w2n

where s is the Laplace operator, nx is the number of modes in x direction, and u>nxh Q , axi, pxi are the natural frequency, structural damping ratio and modal coefficients obtained from modal analysis for each mode (i), respectively. The continuous time domain transfer function [Eq. (10)] can be transformed into discrete time domain using a bilinear transformation, i.e. s = [2(1 - 4 _ 1 )]/[A?(1 + q~1)] where At is the discrete time interval for digital integration and q~l is the backward time shift operator, i.e. q~lx'(t) = x'(t — At). The resulting difference equation for dynamic displacement x' at time interval t becomes, nx

x'(t)

= X [-a,jx'(t + buFxi(t

- Af) - a2ix'(t

- 2Af) +

boiFx(t)

- AO + b2iFxi{t - 2At)]/am

(11)

where a0 = At2uj2mi + 4^L0„xiAt + 4, a, = 2A? 2 wL - 8, a2 = At2u}2„xi - ^tUJnxiAt + 4,

b0 = At2axi + 2At2axi bi = 2At2axi

b2 = At2axl - 2At{3xl

(12)

Similar transformation is used for structural displacements (_y') and cutting forces (Fy) in the normal direction. 4

Simulation and Experimental Results The dynamic milling model has been implemented in C programming language. The input data to the program includes the cutter geometry, workpiece material and dimensions, cutting conditions (axial depth of cut, radial immersion, chip load, 688 / Vol. 120, NOVEMBER 1998

Static Cutting Tests. More than 60 cutting tests were conducted on a vertical CNC milling machine with 30 degrees nominal helix, single fluted, carbide ball end mills for the verification of vibration free, static milling forces first. Slot cutting experiments were chosen on Titanium alloy (Ti6A14V) at different feeds and axial depth of cuts with cutter normal rake angles ranging from 0 to 15 degrees. The feed rate was varied from s, = 0.0127 mm/tooth to 0.127 mm/tooth, the axial depth of cut range was a = 1.27 to a = 0.SR0, and cutters with varying ball radius were tested. The tests were conducted without lubricant and the forces were measured with a three-component Kistler table dynamometer. Using the oblique mechanics model with classical orthogonal cutting machining parameters, the simulation and experimental results were compared. The orthogonal cutting parameters for the Titanium alloy, and its general oblique transformation model to evaluate cutting force coefficients for each oblique cutting edge geometry can be found in Budak et al. (1996) and in Appendix B. Each ball end mill segment is considered to be an oblique cutting edge for the transformation. The percentage difference between the predicted and experimental results has been used to evaluate the statistical reliability of the mechanics model. Mean deviation of —0.93 percent for Fx, 1.36 percent for Fy, and 3.03 percent for Fz were observed from the comparison of 60 ball end milling experiments and simulations. In Fig. 4, a sample simulation of cutting forces for a slotting test at two extreme axial depth of cuts a = 1.27 mm and 6.35 mm at feed-rate 0.0508 mm/tooth are shown. The measured and simulated cutting forces are in good agreement, which proves that the kinematics model of chip thickness for ball end milling and the oblique transformation method used in calculating the milling force coefficients from general orthogonal cutting parameters are sufficiently accurate for the analysis of chatter free mechanics of ball end milling. The correctness of this model is essential in building a reliable dynamic ball end milling model which has increased complexity due to regenerative vibrations. Dynamic Cutting Tests. The performance of the model in predicting the dynamic milling stability is illustrated by separate simulations and experiments conducted on a CNC machining center. A two fluted ball end cutter with a radius of 12.7 mm, 0 degree helix and zero degree rake angle was used in a slotting operation with the same (Ti6A14V) workpiece. The workpiecedynamometer couple had a measured bandwidth of 900 Hz which is well beyond the dominant cutter-spindle modes (452, 514, 556 Hz) and chatter frequency range (720 Hz). The dynamic characteristics of the cutter attached to the spindle are identified from modal tests and shown in table 1. A range of spindle speeds and axial depth of cuts were selected. At each spindle speed, all the cutting conditions except the axial depth of cut were kept constant. The axial depth of cuts were increased until the onset of chatter was detected. The procedure was then repeated for a new spindle speed. In the experiments, chatter onset was determined by observing the output of the cutting forces, as well as the sound pressure recorded by the microphone. A small chip load was chosen in all the tests to prolong the tool life, and each tool was monitored closely throughout the tests and damaged tools were replaced. The experimental and simulation results for chatter stability lobes are presented Transactions of the ASME


Experiments Simulations

200

Axial Depth of Cut -1.27mm Feed » 0.0508 mm/tooth Speed - 269 RPM Rotation Angle [deg]

(4 a)

200

Axial Depth of Cut - 6,35mm Feed - 0.0508 mm/tooth Speed - 269 RPM Rotation Angle [deg]

(4 b)

Fig. 4 Measured and predicted cutting forces for slot cutting tests. Cutting conditions: rake angle = 0 deg, i 0 = 30 deg, N = 269 rev/min, feed rate - 0.0508 mm/rev/tooth, W, = 1 flute, R0 = 9.525 mm. Workpiece material is Titanium alloy Ti6AI4V. See Appendix B for cutting coefficients.

in Fig. 5a. Fine step size was chosen in the simulations: the minimum step size of the axial depth of cut was 0.0125 mm and the spindle speed range was 1000 to 10000 RPM. At low speed range, there are many number of vibration waves left in the immersion zone, therefore the stability lobes are packed and stable pockets are rather small. Also, the process damping becomes dominant and increases the stability at the speeds lower than 1000 rpm. Note that the process damping is usually identified from actual experiments via empirical identification techniques (Tlusty, 1986), and not evaluated here. As the depth of cut increases, the onset of chatter was recognized and recorded, as indicated in the stability chart. The agreement between the chatter behavior of the milling tests and simulations is quite

reasonable. At N = 5000 rev/min, the simulations indicated that the amplitude of resultant cutting forces are about 230 N, and the ball end milling system is relatively stable with negligible vibrations, see Fig. 5b and 5c. When the cutting conditions are moved to an unstable region by selecting a spindle speed of 6000 rpm, the amplitude of resultant cutting forces is more than doubled, although the chip loads (i.e. depth of cut and feed per tooth) are kept constant in both cases. It can be seen from Fig. 5b and 5 c that the predicted and experimentally measured resultant cutting forces are in good agreement. The frequency spectrum of cutting forces also indicate that the dominant frequency is at tooth passing frequency (167 Hz) in stable cut (n = 5000 rpm), and at the chatter frequency (719 Hz) in

Table 1 Time domain simulation conditions for dynamic ball end milling Workpiece material Milling mode Spindle speeds (N) Feed rate (st) Cutter Ball radius (-Ro) Rake angle ( Q „ ) Helix angle (i 0 ) Number of flutes Modal parameters Mode 1 in x Mode 2 in x Mode in y

Titanium alloy Ti6AI4V Slotting 1000 - 10000 [rev/min] 0.0508 mm/tooth DAPRA carbide Ball Nose GWR 25 160 1000RZ Insert - BNNR 1000N FF1 12.7 mm 0 degree 0 degree 2 Force [N], Displacement [m], u>„ [Hz] u w = 452.01, d = 0.0343,0,1 = -0.1819, /J Bl = 1.0266 • 10" s u„,i = 556.11, Cs = 0.0633, a l 2 = -0.4529, /?„, = 2.0606 • 10~5 w„y = 514.60,