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Time domain model of plunge milling operation
ARTICLE IN PRESS
International Journal of Machine Tools & Manufacture 47 (2007) 1351–1361 www.elsevier.com/locate/ijmactool
Time domain model of plunge milling operation Jeong Hoon Ko, Yusuf Altintas The University of British Columbia, Department of Mechanical Engineering, 2054-6250 Applied Science Lane, Vancouver, BC Canada V6T 1Z4 Received 29 June 2006; received in revised form 22 July 2006; accepted 5 August 2006 Available online 27 September 2006
Abstract Plunge milling operations are used to remove excess material rapidly in roughing operations. The cutter is fed in the direction of spindle axis which has the highest structural rigidity. This paper presents time domain modeling of mechanics and dynamics of plunge milling process. The cutter is assumed to be flexible in lateral, axial, and torsional directions. The rigid body feed motion of the cutter and structural vibrations of the tool are combined to evaluate time varying dynamic chip load distribution along the cutting edge. The cutting forces in lateral and axial directions and torque are predicted by considering the feed, radial engagement, tool geometry, spindle speed, and the regeneration of the chip load due to vibrations. The mathematical model is experimentally validated by comparing simulated forces and vibrations against measurements collected from plunge milling tests. The study shows that the lateral forces and vibrations exist only if the inserts are not symmetric, and the primary source of chatter is the torsional–axial vibrations of the plunge mill. The chatter vibrations can be reduced by increasing the torsional stiffness with strengthened flute cavities. r 2006 Elsevier Ltd. All rights reserved. Keywords: Plunge milling; Chatter; Cutting force
1. Introduction The types of plunge milling operations are illustrated in Fig. 1. Since the feed axis coincides with the most rigid spindle axis direction, the process tends to be more vibration free than in plane milling operations. As a result, plunge milling operations have recently become popular in the roughing of cavities in die, mold, and aerospace machining industry. However, it is important to predict the thrust loads and torque as a function of material properties and cutter geometry. The predicted loads are used to size spindle bearings, spindle dimensions, and motor capacity as well as to avoid overloading the machine tool system. There has been limited literature on the plunge milling process, and most efforts have been spent on the design of cutter geometry. Wakaoka et al. [1] studied the intermittent plunge milling process to make vertical walls with high Corresponding author. Tel.: +604 822 5622; fax: +604 822 2403.
E-mail addresses: [email protected] (J.H. Ko), altintas@mech. ubc.ca (Y. Altintas). 0890-6955/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2006.08.007
accuracy and high speed by focusing on the tool geometry and motion. Li et al. [2] presented plunge milling method to create complex chamfer patterns, and estimated cutting forces while neglecting the structural dynamics of the system. The remaining literature belongs to the commercial tool catalogs which present only the dimensions and shape of the plunge milling cutters. A comprehensive mechanics and dynamics of plunge milling, which allows prediction of cutting forces, torque and vibrations, is presented here for the first time to the best knowledge of the authors. A dynamic model allows for the prediction of the performance of plunge milling operations, which helps the optimal design of tools, tool holders, spindles and identification of cutting conditions. There has been significant research done on the mechanics and dynamics of plane milling operations. Tlusty and Ismail [3] presented the time domain simulation of helical end mills by including the structural dynamics of the system. Montgomery and Altintas [4] presented end milling models which consider the integrated rigid body kinematics and structural vibrations. The mechanics of ball end mills, tapered helical ball end mills, and serrated
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Nomenclature D1, D2 shank and effective cutting diameters of the cutter, respectively af, cr, Clf, fp axial rake, cutting edge, axial relief angles of the tool, and cutter pitch angle, respectively yr, Dyr absolute and incremental rigid body rotation angle of the cutter, respectively O spindle speed a radial depth of cut Da length of edge element along radial direction fp pitch angle N the number of flutes C feed per tooth Ft, Fa, Ff, Ty tangential force, radial force, feed force, torque h uncut chip thickness hm possible uncut chip thickness m index of previous tooth passing period T(f) transformation matrix x, y, z, y displacements determined by vibrations
cutters are also studied extensively by researchers using time domain models of the process [5–8]. Plunge milling resembles drilling and boring with inserted heads which are studied by several researchers. Zhang and Kapoor [9] proposed dynamic single-point boring process model which considered the structural flexibility of the boring bar. Lazoglu et al. [10] presented a comprehensive boring bar model which includes the vibration, regeneration of chip thickness, and true kinematics of the boring operation. Atabey et al. [11] presented mechanics of boring heads with multiple inserts. This paper presents an extensive time domain simulation model of plunge milling, which is organized as follows. The mechanics of rigid plunge milling without the presence of vibrations are presented in Section 2. The true kinematics of plunge milling, where the tool rotates while feeding into the material, are considered. Section 3 presents the
Xc, Yc, Zc, yc present cutter center position determined by rigid body motion and vibration (Xcm, Ycm, Zcm, ycm) previous cutter center position i, j, k Tooth index, cutter rotation angle index, and radial disk element index (Xe, Ye, Ze, ye) present cutting edge position (Xem, Yem, Zem, yem) previous cutting edge position Kt, Ka, Kf tangential, radial, and frictional cutting force coefficients yv angular variation rk radial distance of the present edge element ld distance between the previous cutter center and the present edge position Fab frequency response function of the structure onh, kh, and zh natural frequency, modal stiffness, and damping ratio Rr, Ra radial and axial run-out jm previous cutter rotation angle index iprev tooth index at previous cutter rotation angle hm possible dynamic uncut chip thicknesses
influence of structural vibrations and regeneration of chip thickness on dynamic cutting forces. The time domain simulation results are experimentally verified in Section 4. The paper is concluded with a summary of contributions of the time domain solution model of plunge milling. 2. Modeling of plunge milling mechanics The geometry and parameters of a plunge milling cutter are shown in Fig. 2. The cutter cuts the metal at the bottom cutting edges of the inserts as it plunges into the metal in spindle axis (z) direction. The inserts have an offset distance of l from the cutter center. The tangential cutting force (Ft) is in the direction of cutting speed and distributed along the cutting edge at the bottom. The feed force (Ff) is in the direction of feed or z-axis, and the passive force (Fa) acts along the cutting edge. As the cutting point
Fig. 1. Plunge milling process configuration: (a) plunge milling process for making large hole, (b) plunge milling process to enlarge a hole and (c) intermittent plunge milling process to make vertical wall or conduct rough cutting.
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described by fði; j; kÞ ¼ yr ðjÞ þ ifp ;
yr ¼ j Dyr ¼ Ot,
(1)
where yr is the rotation angle of the cutter measured from y axis, fp ¼ 2p/N is the pitch angle and N is the number of teeth on the cutter. The differential area of chip cut is dAc ¼ h(f)Da where h(f) is the instantaneous chip thickness removed by a cutting edge segment with a differential length of Da. The tangential (Ft), radial (Fa), feed forces (Ff), and torque (Ty) acting on the differential cutting edge element are given by dF t ¼ K t dAc ; dF f ¼ K f dAc ;
dF a ¼ K a dAc ;
(2)
dT y ¼ ðK t dAc Þrk ;
where 1 rk ¼ k Da þ l. 2 The tangential, radial, and feed forces defined on the cutting edge can be transformed into three orthogonal force components in Cartesian coordinates of the cutter axes as follows: 2 3 2 3 Fx Ft 6 Fy 7 6 Fa 7 6 7 6 7 (3) 6 7 ¼ TðfÞ6 7, 4 Fz 5 4 Ff 5 Ty Ty where T(f) is the transformation matrix: 2 3 cos f sin f 0 0 6 sin f cos f 0 0 7 6 7 TðfÞ ¼ 6 7. 4 0 0 1 05 0 Fig. 2. Geometry and coordinates of a sample plunge mill. Cutter parameters: D1 ¼ 20 mm, D2 ¼ 25 mm, l ¼ 5.5 mm, af ¼ 101, cr ¼ 101, Clf ¼ 51. (Sandvik Cutter Part no.: R210-025A20-09 M).
approaches the cutter center, the cutting speed decreases. Depending on the insert geometry and run-out, the effective rake angle of the cutting edge as well as the elevation in z and position in x, y directions may change, which alters the cutting mechanics. In order to consider a general case, the insert is divided into finite number of small differential elements in the radial direction. The chip load and corresponding differential loads for each edge element are evaluated and digitally integrated to predict the total forces in three directions and torque. The algorithm is repeated at discrete time or spindle rotation intervals to predict the time varying forces in plunge milling. The angular position of the points along the cutting edge of the cutter at its bottom is evaluated in Fig. 2(b). The angular position (fði; j; kÞ) of the kth edge element of the tooth i at the jth angular position of the cutter is
0
0
(4)
1
The total instantaneous cutting forces and torque in Cartesian coordinates can be evaluated by digitally integrating the contributions of all differential edge segments of the inserts as follows when the reference tooth is at an angular position (yr): Fx ¼
N 1 M1 X X
½K t cosðfÞ K a sinðfÞ dAc ,
n¼0 m¼0
Fy ¼
N 1 M1 X X
½K t sinðfÞ K a cosðfÞ dAc ,
ð5Þ
n¼0 m¼0
Fz ¼
N 1 M1 X X n¼0 m¼0
K f dAc ;
Ty ¼
N 1 M1 X X
ðK tc Þrk dAc ,
n¼0 m¼0
where N is the number of teeth on the cutter and M is the number of elements on each cutting edge. As the cutter rotates, the cutting forces are re-evaluated at each discrete angular position of the cutter.
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3. Dynamics of plunge milling The influence of present and past structural vibrations on the chip generation mechanism is considered in modeling the dynamics of plunge milling. The overall model is based on the procedure shown in Fig. 3. The cutter geometry, cutting conditions, insert run-out, modal parameters of the structure’s dynamics at the tool tip, and cutting force coefficients are defined as inputs. The cutter position, and the corresponding chip loads and cutting forces are evaluated using the mechanics of plunge milling presented in the previous section. The cutting forces are applied on the structural model of the system, and the corresponding cutter position due to vibrations and rigid body plunge feeding motion are predicted. The dynamic chip thickness distribution is evaluated by subtracting the present cutting edge position from the previous tooth location which has one tooth period time delay. The procedure is repeated at discrete time intervals which are at least ten times less than the natural period of the highest modal frequency of the structure. 3.1. Equation of dynamic plunge milling It is assumed that the structure experiences two lateral (x, y), axial (z) and torsional (y) vibrations. Also, the coupling between the torsional and axial displacements is considered. The frequency response function of each degree of freedom is measured independently through impact modal tests, and cross-coupling in lateral directions are omitted since they are negligible. Although the structural dynamics equations are expressed as a set of uncoupled differential equations, the vibrations in each direction affect the chip thickness, which in turn affects all cutting forces and makes the system coupled, containing regenerative or time delayed vibration terms. Such time delayed, coupled differential equation systems are best solved numerically in time domain, as explained in the following. The general equation of motion for the overall system is given as follows: 8 9 2 38 9 Fxx 0 0 0 > Fx > x> > > > > > > > = 6 0 = < > Fyy 0 0 7 6 7 Fy ¼6 , (6) 7 > 0 Fzz Fzy 5> F > z> 4 0 > > > > > > > z> ; : ; : 0 0 Fyz Fyy Ty y
where (x, y, z, y) represent the linear and torsional vibrations produced by cutting forces (Fx, Fy, Fz) and torque (Ty), respectively. The direct (Fxx, Fyy, Fzz,Fyy) and cross (Fzy, Fyz) transfer functions of the plunge mill at its tip are expressed by their modal parameters as Fab ðsÞ ¼
H o2nh =kh Da X ¼ , F b h¼1 s2 þ 2zh onh s þ o2nh
(7)
where ab ( a or b is x, y, z, or y) denotes the displacement of the cutter center coordinate a when cutting force F is applied in direction b. H is the total number of modes in the system, h represents each of these modes and onh, kh, and zh are the natural frequency, modal stiffness, and damping ratio, respectively. The axial and torsional vibrations are coupled in this particular cutter, which are represented by crosstransfer functions (Fzy, Fyz). Note that while the plunge mills are always flexible in lateral directions, the contributions of axial, torsional, and coupled torsional–axial modes very much depend on the geometry of cavities between the teeth. The frequency response functions in rigid directions can be set to zero in Eq. (6) which represents a general case. The fourth-order Runge–Kutta method is used for the time domain, numerical integration of the differential Eq. (6) which leads to the prediction of vibrations for the applied cutting forces and torque on the plunge mill. The total displacement of the tool is evaluated by summing the vibrations contributed by all natural modes: DaðtÞ ¼
H X
(8)
Dah ðtÞ,
h¼1
The dynamic chip load distribution along the cutting edge is evaluated by predicting the present and previous cutting edges positions on the chip surface, which requires tracking the cutter center position of the plunge mill (Xc, Yc, Zc, yc), which can be described using the vibration displacement and feed per tooth (ft) as follows: 9 9 8 8 9 8 X c ðtÞ > xðtÞ > > > Rr sin f > > > > > > > > > > > > > > > = = = < Y c ðtÞ > < yðtÞ > < R cos f > r ¼ þ , (9) NcOt Z c ðtÞ > zðtÞ > > > > > > > > > > > 2p > > > > > > > > ; > > : : : yc ðtÞ ; yðtÞ ; Ot |fflfflfflfflfflfflfflfflffl ffl {zfflfflfflfflfflfflfflfflffl ffl } |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl} CUTTER CENTER
RIGID BODY
VIBRATIONS
where Rr represents the radial eccentricity of the cutter center. The cutter center history is stored in an array as a function of cutter rotation angle index (j) or time, which is used for dynamic uncut chip thickness calculation. 3.2. Dynamic uncut chip thickness model
Fig. 3. Time domain simulation procedure for dynamic plunge milling.
Montgomery and Altintas [4] developed an exact kinematic model of dynamic milling by digitizing the surface finish at discrete time intervals. Although their model accurately predicts the exact chip thickness with regenerative vibration effects in milling process, it is
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Fig. 4. Dynamic uncut chip thickness model in plunge milling.
computationally costly and not robust when torsional vibrations are included. An alternative but still an accurate kinematic model of dynamic chip thickness is formulated here by considering the structural vibrations in the lateral (x, y), axial (z), and torsional (y) directions at the cutter center. As shown in Fig. 4, the dynamic chip thickness changes with time by reflecting the regenerative vibration marks left on either side of the chip surface by the present and previous teeth in plunge milling. When the present and previous cutting edge positions are at the same position (Xe ¼ Xem, Ye ¼ Yem) in XY plane, the dynamic chip thickness is defined as the difference between z position (Ze) of the present and z position (Zem) of the previous cutting edge positions. The edge positions ((Xe, Ye, Ze) and (Xem, Yem, Zem)) are calculated using the present and previous cutter center positions ((Xc, Yc, Zc), (Xcm, Ycm, Zcm)) and the cutter geometry information. Since the cutter center history is updated and saved in an array as a function of the cutter rotation angle index (j) or simulation time interval, the dynamic chip load produced by the present and previous teeth can be evaluated easily. The method of tracking the previous cutting edge position is explained in the following. As the angular position (fem) of the previous cutting edge approaches (fe) of the present cutting edge with respect to previous cutter center position (Xcm, Ycm), the previous (Xem, Yem) and the present cutting edge positions (Xe, Ye) become the same. The present and previous cutting edge positions can be evaluated from the time history of the cutter center when fem is equal to fe. The angular position (fm) of the previous cutting edge is located from the algorithm presented in Fig. 5. First, the initial previous cutter rotation angle index (jm) is estimated in order to
locate the previous angular position of the cutter by rotating the cutter backward by a pitch angle: j m ¼ j mj p
ðm ¼ 1; 2; . . .Þ,
(10)
where jp ¼ fp/Dyr and m is the index of the previous tooth passing period. The previous cutter center information (Xcm, Ycm, Zcm, ycm) is extracted according to the index jm. Using the previous center positions, two location angles (fem, fe) of the present and the previous cutting edges are evaluated as follows: fe ¼ tan
ðX e X cm Þ ; ðY e Y cm Þ
fem ¼ iprev fp þ ycm ,
(11)
where iprev is calculated as the remainder of i+m/N, which indicates the tooth index at the previous cutter rotation angle. The lateral and torsional vibrations cause the difference (yv) between two angular positions (fem, fe): yv ¼ fe fem ¼ tan
ðX e X cm Þ fem , ðY e Y cm Þ
(12)
when yv approaches zero, two angles fem and fe as well as two cutting edge positions (Xe, Ye) and (Xem, Yem) match at the present cutting edge element. If yv is not zero, jm is modified using Eq. (13) and the procedure is applied until yv becomes zero: jm ¼ jm þ
yv . Dyr
(13)
The dynamic chip thickness is calculated using Z positions of the present and the previous cutting edges when yv approaches zero. First the present cutting edge
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The dynamic uncut chip thickness can be calculated as follows: hm ði; j; k; mÞ ¼ Z em Z e ,
(16)
The same procedure is applied for calculating the possible dynamic uncut chip thicknesses (hm) which could be generated by the present and the previous teeth at the previous tooth passing periods (m ¼ 1; 2; 3 . . .). The minimum of the possible dynamic uncut chip thicknesses is selected as the real dynamic chip thickness. If the dynamic chip load is less than zero, the tool is assumed to jump out of cut in order to account for the nonlinearity in the process: hði; j; kÞ ¼ max½0; minðhm ði; j; k; mÞÞ.
(17)
The evaluated dynamic uncut chip thickness is applied on the tool tip according to the force expression given in Eq. (5). 4. Experimental validation A series of plunge milling tests have been conducted to validate the dynamic plunge milling model presented in the article. 4.1. Mechanistic identification of cutting force coefficients
Fig. 5. The algorithm to evaluate dynamic chip thickness. (iprev indicates the tooth index at previous cutter rotation angle; jm is previous cutter rotation angle index; hm denotes the possible dynamic chip thicknesses).
The accuracy of predicting the cutting forces and resulting vibrations are highly dependent on the accurate identification of cutting force coefficients. Since the chip load per tooth is small and the insert geometry can be too complex in plunge milling, the cutting coefficients are identified as nonlinear functions of the chip thickness as follows. The intermittent plunge milling tests, (Fig. 1(c)) are used to identify cutting force coefficients. The cutting force equations given in Eq. (5), are re-written in matrix form as 9 8 9 2 38 A11 A12 A13 > > = = < Fx > < Kt > 7 Fy ¼ 6 (18) 4 A21 A22 A23 5 K a , > > > ; ; :F > : A31 A32 A33 Kf z where
location is calculated from the cutter center position as shown below:
A11 ¼
ð14Þ
where Rai is the axial run-out of ith tooth, and Xc, Yc, Zc, and yc are obtained from Eq. (9). The Z position of the previous cutting edge is calculated as follows: Z em ¼ Zcm þ l d tan cr þ Raiprev , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l d ¼ ðX e X cm Þ2 þ ðY e Y cm Þ2 .
ð15Þ
ð cos fÞ hðfÞ Da,
n¼0 k¼0
X e ¼ X c þ rk sinðyc þ ifp Þ; Y e ¼ Y c þ rk cosðyc þ ifp Þ, Z e ¼ Zc þ Rai þ ðrk lÞ tan cr ,
N 1 N k 1 X X
A12 ¼
N 1 N k 1 X X
ðsin fÞhðfÞ Da;
A13 ¼ 0,
n¼0 k¼0
A21 ¼
N 1 N k 1 X X
ðsin fÞhðfÞ Da,
n¼0 k¼0
A22 ¼
N 1 N k 1 X X
ðcos fÞhðfÞ Da;
n¼0 k¼0
A23 ¼ 0,
ARTICLE IN PRESS J.H. Ko, Y. Altintas / International Journal of Machine Tools & Manufacture 47 (2007) 1351–1361 Table 1 Identified cutting coefficients in plunge milling of aluminum alloy Al7050T7451, see Fig. 2 for cutter geometry Kt (kN/mm2)
Ka (kN/mm2)
Kf (kN/mm2)
C1 ¼ 9.887 C2 ¼ 0.796 C3 ¼ 0.358 C4 ¼ 0.769
C1 ¼ 1.662 C2 ¼ 0.652 C3 ¼ 4.186 C4 ¼ 3.375
C1 ¼ 4.095 C2 ¼ 0.721 C3 ¼ 2.723 C4 ¼ 1.577
A31 ¼ 0;
A32 ¼ 0;
A33 ¼
N1 k 1 X NX
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hðfÞ Da.
n¼0 k¼0
The cutting forces are measured during the chatter vibration free plunge milling tests, and geometric parameters (A) are evaluated at each time or angular position of the cutter. The cutting coefficients (Kt, Ka, and Kf) are evaluated by applying the least squares to Eq. (19). The nonlinearity in the cutting coefficients are
Table 2 Modal parameters of the plunge mill attached to Mori Seiki SH403 with HSK 63 interface Mode #
Natural frequency (Hz)
Damping ratio (z)
Modal stiffness (k)
Dynamic stiffness (2 kz)
xx
1 2 3
508 3040 4035
0.0235 0.0242 0.0179
121.00 N/mm 41.10 N/mm 67.50 N/mm
5.68 N/mm 1.99 N/mm 2.42 N/mm
yy
1 2 3
515 3100 3961
0.0510 0.0305 0.0183
44.30 N/mm 80.40 N/mm 51.00 N/mm
4.52 N/mm 4.90 N/mm 1.87 N/mm
zz
1 2
321 405
0.0482 0.0379
242.00 N/mm 790.00 N/mm
23.30 N/mm 59.80 N/mm
zy
1
12090
0.00153
21.20 Nm/mm
0.0649 Nm/mm
yy
1
12000
0.00243
1.106E+04 Nm/rad
53.80 Nm/rad
Fig. 6. Comparison of the predicted and measured cutting forces under chatter-free cutting conditions. Work material: Al7050-T7451, Cutter: see Fig. 2. (a) The predicted and measured cutting forces for two revolutions under fully immersed plunge milling. (a-1) Cutting conditions: spindle speed ¼ 1000 rpm, feed per tooth ¼ 0.075 mm/tooth, radial depth of cut ¼ 2.5 mm. (a-2) Cutting conditions: spindle speed ¼ 1000 rpm, feed per tooth ¼ 0.075 mm/tooth, radial depth of cut ¼ 6.5 mm. (b) The predicted cutting forces and the measured ones for two revolutions under intermittent plunge milling. (b-1) Cutting conditions: spindle speed ¼ 1000 rpm, feed per tooth ¼ 0.05 mm/tooth, radial depth of cut ¼ 4 mm. (b-2) Cutting conditions: spindle speed ¼ 1000 rpm, feed per tooth ¼ 0.125 mm/tooth, radial depth of cut ¼ 3 mm.
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represented by C1 C2 þ C2, (19) 1 þ ðh=C 3 ÞC 4 which indicates that the coefficients have high values at low chip loads, but approaches to a constant value at large chip loads [7]. The identified cutting coefficients for plunge milling of Aluminum alloy is given in Table 1.
K¼
4.2. Plunge milling tests A series of plunge milling tests were conducted at various radial immersion conditions to validate the process model. The structural modal parameters of the plunge milling tool used in the experiments are given in Table 2. While experimental modal tests for low frequencies were performed with Kistler hammer 9722, a PCB 086D80 miniature impulse hammer was used for modes with frequencies higher than 1.5 kHz in order to expand the frequency bandwidth of the impact excitation. Kistler
accelerometer model 8778A00 was attached to the plunge mill to measure the vibrations. The work material was Al7050-T7451 and the cutter geometry is given in Fig. 2. One insert had 20 mm axial and 1.5 mm radial measured run-outs relative to the reference insert, which are included in the simulation model. The machine tool was a Mori Seiki SH403 Horizontal Machining Center, and a Sandvik CG20 HSK63 holder was used during cutting tests. Sample plunge milling test results at n ¼ 1000 rpm spindle speed but at various immersion and feed rates are given in Fig. 6, where the process was free of chatter vibrations. The predicted and measured cutting forces are in good agreement in all cases, which indicates the correctness of rigid body kinematics of plunge milling, chip thickness evaluation from digitized surfaces cut by successive teeth, and nonlinear cutting coefficient model. The figures show that when the cutter is fully immersed, like in drilling (see Fig. 1(b)), the feed force in plunge direction (Fz) is constant, and in plane cutting force (Fx, Fy) amplitudes are small due to cancellation of forces
Fig. 7. Analysis of measured cutting force in case of chatter. Cutting conditions: spindle speed ¼ 16000 rpm, feed per tooth ¼ 0.075 mm/tooth, radial depth of cut ¼ 5 mm. Work material: Al7050-T7451, Cutter: see Fig. 2. (a) Measured cutting forces Fx and Fy for one revolution. (b) Measured cutting force Fz. (c) FFT of the measured Fx. (d) FFT of the measured Fy. (e) FFT of the measured Fz. (f) FFT of the measured sound.
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generated by two symmetric inserts, see Fig. 6(a). The oscillation in lateral forces (Fx, Fy) is due mainly to run-out which upsets the chip load balance between the two inserts. When the radial immersion is intermittent as illustrated in Fig. 1(c), the cutting forces become periodic at tooth passing intervals, see Fig. 6(b), due to loss of force cancellation. Since the accuracy of the simulation model is verified in chatter-free cutting conditions, the proposed dynamic model can be relied on in predicting dynamic milling more comfortably. Fig. 7 shows the measured cutting forces, Fourier spectrum of cutting forces and sound record when the system chattered at the spindle speed of 16,000 rpm with feed rate of 0.075 mm/tooth during a full immersion plunge milling test. Although the axial force Fz should be constant in full immersion cut, it fluctuates at 12360 Hz chatter frequency which is caused by the coupled torsional–axial mode of the plunge mill, see Table 2. The spectrum of the
1359
sound shows harmonics which are spread on either side of the chatter frequency at tooth passing frequency (533.3 Hz) intervals [12]. The simulation of the same cutting conditions is shown in Fig. 8, which shows the growth of chatter vibrations within ten spindle revolutions. Similar to the measured values, the predicted plunge milling forces clearly indicate the chatter occurring at 12137 Hz. The small discrepancy between the simulated and measured chatter frequency is mainly due to resolution of FFT and numerical integration of the delayed differential equations in evaluating the chip thickness. The high-frequency component is generated by torsional–axial vibration coupling effect, where the axial displacement (zy) is due to the applied torque as shown in Table 2. Unlike cutting forces Fx and Fy, torque Fy generated by each tooth is not canceled and influences the axial displacement by the modal parameter zy. Here, the torsional vibrations are
Fig. 8. Analysis of simulation results in case of chatter. Cutting conditions: same as in Fig. 7. (a) Predicted cutting forces for 10 revolutions. (b) Predicted cutting force for one revolution. (c) Predicted torsion force Ty. (d) Cutter deflection in Z direction. (e) FFT of the predicted Fx. (d) FFT of the predicted Fz.
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Fig. 9. The solid model of the plunge mill, and its axial deformation when torsion is applied to cutting edge in FE model.
translated as axial displacements of the cutter which directly affect the dynamic chip thickness. Finite Element model of the plunge mill is constructed to make sure the dominant mode is due to torsional–axial coupling as shown in Fig. 9, which gives a similar natural frequency (around 12,600 Hz) and axial displacement when torsion is applied. Although the axial modes, the influence of axial mode on torsional displacements, as well as the pure torsional mode are given in Table 2 for completeness, they have very little influence on the chatter stability in this particular set-up. When the speed is increased to 17,142 rpm, the measured and predicted cutting forces are compared in Fig. 10. Stable cutting is found in both simulation and experiment at this cutting condition and the predicted cutting forces are in good agreement with the measured ones. It is verified that stable cutting condition, which leads to high productivity can be pursued using this time domain solution, and that process can also be accurately predicted prior to real machining. 5. Conclusions The time domain model of the dynamic plunge milling process allows prediction of cutting forces, torque, power, and vibrations in plunge milling operations as a function of tool geometry, structural dynamics of the machine and cutting conditions. The cutting forces in plane milling act in lateral directions, hence the major vibrations are dominated by
Fig. 10. Comparison of the measured and predicted cutting forces under stable cutting condition with high spindle speed. Cutting conditions: spindle speed ¼ 17142 rpm, feed per tooth ¼ 0.075 mm/tooth, radial depth of cut ¼ 5 mm. Work material: Al7050-T7451, Cutter: see Fig. 2. (a) The measured cutting forces. (b) The predicted cutting forces.
the bending modes of the tool–holder–spindle assembly. However, the paper shows that the major cutting forces act in the torsional and cutter-spindle axis directions in plunge milling. The lateral forces can be canceled by using symmetric teeth on the plunge mills. While the spindle axis is usually most rigid, the chip evacuation cavities weaken the cross-section of the plunge mills which lead to torsional flexibility in plunge mills. The torsional vibrations are transmitted as axial displacements through structural coupling, and affect the chip thickness regeneration most, hence causing chatter. The torsional chatter can be reduced by designing the plunge mills with strengthened chip evacuation cavities. The lateral vibrations can be reduced by minimizing the tool run-outs and using symmetric tools.
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Acknowledgments This work is supported by the Post-doctoral Fellowship Program of Korea Research Foundation (KRF) for the lead author, and by the National Sciences and Engineering Research Council of Canada (NSERC) and Pratt and Whitney Canada under Industrial Research Chair and Strategic Grants of the second author. The cutting tools and tool holders are provided by Mitsubishi Materials and Sandvik Coromant Companies, and the high speed Mori Seiki SH403 horizontal machining center is contributed by Machine Tool technology Research Foundation (MTTRF) Company. References [1] S. Wakaoka, Y. Yamane, K. Sekiya, N. Narutaki, High-speed and high-accuracy plunge cutting for vertical walls, J. Mater. Process. Technol. 127 (2002) 246–250. [2] Y. Li, S.Y. Liang, R.C. Petrof, B.B. Seth, Force modelling for cylindrical plunge cutting, Int. J. Adv. Manuf. Technol. 16 (2004) 863–870. [3] J. Tlusty, F. Ismail, Basic nonlinearity in machining chatter, CIRP Ann. 30 (1981) 21–25.
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[4] D. Montgomery, Y. Altintas, Mechanism of cutting force and surface generation in dynamic milling, ASME J. Eng. Ind. 113 (2) (1991) 160–168. [5] Y. Altintas, P. Lee, Mechanics and dynamics of ball end milling, ASME J. Manuf. Sci. Eng. 120 (1998) 684–692. [6] Y. Altintas, P. Lee, A general mechanics and dynamics model for helical end mills, CIRP Ann. 45 (1996) 59–64. [7] J.H. Ko, D.-W. Cho, 3D ball-end milling force model using instantaneous cutting force coefficients, ASME J. Manuf. Sci. Eng. 127 (2005) 1–12. [8] S.E. Merdol, Y. Altintas, Mechanics and dynamics of serrated cylindrical and tapered end mills, ASME J. Manuf. Sci. Eng. 126 (2004) 318–326. [9] G.M. Zhang, S.G. Kapoor, Dynamic modeling and analysis of the boring machining system, ASME J. Eng. Ind. 109 (1987) 219–226. [10] I. Lazoglu, F. Atabey, Y. Altintas, Dynamics of boring processes: part iii-time domain modeling, Int. J. Mach. Tools. Manuf. 42 (14) (2002) 1567–1576. [11] F. Atabey, I. Lazoglu, Y. Altintas, Mechanics of boring processes: Part II—multi-insert boring heads, Int. J. Mach. Tools. Manuf. 43 (5) (2002) 477–484. [12] Y. Altintas, Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design, Cambridge University Press. ISBN 0-521-65973-6, 2000.
International Journal of Machine Tools & Manufacture 47 (2007) 1351–1361 www.elsevier.com/locate/ijmactool
Time domain model of plunge milling operation Jeong Hoon Ko, Yusuf Altintas The University of British Columbia, Department of Mechanical Engineering, 2054-6250 Applied Science Lane, Vancouver, BC Canada V6T 1Z4 Received 29 June 2006; received in revised form 22 July 2006; accepted 5 August 2006 Available online 27 September 2006
Abstract Plunge milling operations are used to remove excess material rapidly in roughing operations. The cutter is fed in the direction of spindle axis which has the highest structural rigidity. This paper presents time domain modeling of mechanics and dynamics of plunge milling process. The cutter is assumed to be flexible in lateral, axial, and torsional directions. The rigid body feed motion of the cutter and structural vibrations of the tool are combined to evaluate time varying dynamic chip load distribution along the cutting edge. The cutting forces in lateral and axial directions and torque are predicted by considering the feed, radial engagement, tool geometry, spindle speed, and the regeneration of the chip load due to vibrations. The mathematical model is experimentally validated by comparing simulated forces and vibrations against measurements collected from plunge milling tests. The study shows that the lateral forces and vibrations exist only if the inserts are not symmetric, and the primary source of chatter is the torsional–axial vibrations of the plunge mill. The chatter vibrations can be reduced by increasing the torsional stiffness with strengthened flute cavities. r 2006 Elsevier Ltd. All rights reserved. Keywords: Plunge milling; Chatter; Cutting force
1. Introduction The types of plunge milling operations are illustrated in Fig. 1. Since the feed axis coincides with the most rigid spindle axis direction, the process tends to be more vibration free than in plane milling operations. As a result, plunge milling operations have recently become popular in the roughing of cavities in die, mold, and aerospace machining industry. However, it is important to predict the thrust loads and torque as a function of material properties and cutter geometry. The predicted loads are used to size spindle bearings, spindle dimensions, and motor capacity as well as to avoid overloading the machine tool system. There has been limited literature on the plunge milling process, and most efforts have been spent on the design of cutter geometry. Wakaoka et al. [1] studied the intermittent plunge milling process to make vertical walls with high Corresponding author. Tel.: +604 822 5622; fax: +604 822 2403.
E-mail addresses: [email protected] (J.H. Ko), altintas@mech. ubc.ca (Y. Altintas). 0890-6955/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2006.08.007
accuracy and high speed by focusing on the tool geometry and motion. Li et al. [2] presented plunge milling method to create complex chamfer patterns, and estimated cutting forces while neglecting the structural dynamics of the system. The remaining literature belongs to the commercial tool catalogs which present only the dimensions and shape of the plunge milling cutters. A comprehensive mechanics and dynamics of plunge milling, which allows prediction of cutting forces, torque and vibrations, is presented here for the first time to the best knowledge of the authors. A dynamic model allows for the prediction of the performance of plunge milling operations, which helps the optimal design of tools, tool holders, spindles and identification of cutting conditions. There has been significant research done on the mechanics and dynamics of plane milling operations. Tlusty and Ismail [3] presented the time domain simulation of helical end mills by including the structural dynamics of the system. Montgomery and Altintas [4] presented end milling models which consider the integrated rigid body kinematics and structural vibrations. The mechanics of ball end mills, tapered helical ball end mills, and serrated
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Nomenclature D1, D2 shank and effective cutting diameters of the cutter, respectively af, cr, Clf, fp axial rake, cutting edge, axial relief angles of the tool, and cutter pitch angle, respectively yr, Dyr absolute and incremental rigid body rotation angle of the cutter, respectively O spindle speed a radial depth of cut Da length of edge element along radial direction fp pitch angle N the number of flutes C feed per tooth Ft, Fa, Ff, Ty tangential force, radial force, feed force, torque h uncut chip thickness hm possible uncut chip thickness m index of previous tooth passing period T(f) transformation matrix x, y, z, y displacements determined by vibrations
cutters are also studied extensively by researchers using time domain models of the process [5–8]. Plunge milling resembles drilling and boring with inserted heads which are studied by several researchers. Zhang and Kapoor [9] proposed dynamic single-point boring process model which considered the structural flexibility of the boring bar. Lazoglu et al. [10] presented a comprehensive boring bar model which includes the vibration, regeneration of chip thickness, and true kinematics of the boring operation. Atabey et al. [11] presented mechanics of boring heads with multiple inserts. This paper presents an extensive time domain simulation model of plunge milling, which is organized as follows. The mechanics of rigid plunge milling without the presence of vibrations are presented in Section 2. The true kinematics of plunge milling, where the tool rotates while feeding into the material, are considered. Section 3 presents the
Xc, Yc, Zc, yc present cutter center position determined by rigid body motion and vibration (Xcm, Ycm, Zcm, ycm) previous cutter center position i, j, k Tooth index, cutter rotation angle index, and radial disk element index (Xe, Ye, Ze, ye) present cutting edge position (Xem, Yem, Zem, yem) previous cutting edge position Kt, Ka, Kf tangential, radial, and frictional cutting force coefficients yv angular variation rk radial distance of the present edge element ld distance between the previous cutter center and the present edge position Fab frequency response function of the structure onh, kh, and zh natural frequency, modal stiffness, and damping ratio Rr, Ra radial and axial run-out jm previous cutter rotation angle index iprev tooth index at previous cutter rotation angle hm possible dynamic uncut chip thicknesses
influence of structural vibrations and regeneration of chip thickness on dynamic cutting forces. The time domain simulation results are experimentally verified in Section 4. The paper is concluded with a summary of contributions of the time domain solution model of plunge milling. 2. Modeling of plunge milling mechanics The geometry and parameters of a plunge milling cutter are shown in Fig. 2. The cutter cuts the metal at the bottom cutting edges of the inserts as it plunges into the metal in spindle axis (z) direction. The inserts have an offset distance of l from the cutter center. The tangential cutting force (Ft) is in the direction of cutting speed and distributed along the cutting edge at the bottom. The feed force (Ff) is in the direction of feed or z-axis, and the passive force (Fa) acts along the cutting edge. As the cutting point
Fig. 1. Plunge milling process configuration: (a) plunge milling process for making large hole, (b) plunge milling process to enlarge a hole and (c) intermittent plunge milling process to make vertical wall or conduct rough cutting.
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described by fði; j; kÞ ¼ yr ðjÞ þ ifp ;
yr ¼ j Dyr ¼ Ot,
(1)
where yr is the rotation angle of the cutter measured from y axis, fp ¼ 2p/N is the pitch angle and N is the number of teeth on the cutter. The differential area of chip cut is dAc ¼ h(f)Da where h(f) is the instantaneous chip thickness removed by a cutting edge segment with a differential length of Da. The tangential (Ft), radial (Fa), feed forces (Ff), and torque (Ty) acting on the differential cutting edge element are given by dF t ¼ K t dAc ; dF f ¼ K f dAc ;
dF a ¼ K a dAc ;
(2)
dT y ¼ ðK t dAc Þrk ;
where 1 rk ¼ k Da þ l. 2 The tangential, radial, and feed forces defined on the cutting edge can be transformed into three orthogonal force components in Cartesian coordinates of the cutter axes as follows: 2 3 2 3 Fx Ft 6 Fy 7 6 Fa 7 6 7 6 7 (3) 6 7 ¼ TðfÞ6 7, 4 Fz 5 4 Ff 5 Ty Ty where T(f) is the transformation matrix: 2 3 cos f sin f 0 0 6 sin f cos f 0 0 7 6 7 TðfÞ ¼ 6 7. 4 0 0 1 05 0 Fig. 2. Geometry and coordinates of a sample plunge mill. Cutter parameters: D1 ¼ 20 mm, D2 ¼ 25 mm, l ¼ 5.5 mm, af ¼ 101, cr ¼ 101, Clf ¼ 51. (Sandvik Cutter Part no.: R210-025A20-09 M).
approaches the cutter center, the cutting speed decreases. Depending on the insert geometry and run-out, the effective rake angle of the cutting edge as well as the elevation in z and position in x, y directions may change, which alters the cutting mechanics. In order to consider a general case, the insert is divided into finite number of small differential elements in the radial direction. The chip load and corresponding differential loads for each edge element are evaluated and digitally integrated to predict the total forces in three directions and torque. The algorithm is repeated at discrete time or spindle rotation intervals to predict the time varying forces in plunge milling. The angular position of the points along the cutting edge of the cutter at its bottom is evaluated in Fig. 2(b). The angular position (fði; j; kÞ) of the kth edge element of the tooth i at the jth angular position of the cutter is
0
0
(4)
1
The total instantaneous cutting forces and torque in Cartesian coordinates can be evaluated by digitally integrating the contributions of all differential edge segments of the inserts as follows when the reference tooth is at an angular position (yr): Fx ¼
N 1 M1 X X
½K t cosðfÞ K a sinðfÞ dAc ,
n¼0 m¼0
Fy ¼
N 1 M1 X X
½K t sinðfÞ K a cosðfÞ dAc ,
ð5Þ
n¼0 m¼0
Fz ¼
N 1 M1 X X n¼0 m¼0
K f dAc ;
Ty ¼
N 1 M1 X X
ðK tc Þrk dAc ,
n¼0 m¼0
where N is the number of teeth on the cutter and M is the number of elements on each cutting edge. As the cutter rotates, the cutting forces are re-evaluated at each discrete angular position of the cutter.
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3. Dynamics of plunge milling The influence of present and past structural vibrations on the chip generation mechanism is considered in modeling the dynamics of plunge milling. The overall model is based on the procedure shown in Fig. 3. The cutter geometry, cutting conditions, insert run-out, modal parameters of the structure’s dynamics at the tool tip, and cutting force coefficients are defined as inputs. The cutter position, and the corresponding chip loads and cutting forces are evaluated using the mechanics of plunge milling presented in the previous section. The cutting forces are applied on the structural model of the system, and the corresponding cutter position due to vibrations and rigid body plunge feeding motion are predicted. The dynamic chip thickness distribution is evaluated by subtracting the present cutting edge position from the previous tooth location which has one tooth period time delay. The procedure is repeated at discrete time intervals which are at least ten times less than the natural period of the highest modal frequency of the structure. 3.1. Equation of dynamic plunge milling It is assumed that the structure experiences two lateral (x, y), axial (z) and torsional (y) vibrations. Also, the coupling between the torsional and axial displacements is considered. The frequency response function of each degree of freedom is measured independently through impact modal tests, and cross-coupling in lateral directions are omitted since they are negligible. Although the structural dynamics equations are expressed as a set of uncoupled differential equations, the vibrations in each direction affect the chip thickness, which in turn affects all cutting forces and makes the system coupled, containing regenerative or time delayed vibration terms. Such time delayed, coupled differential equation systems are best solved numerically in time domain, as explained in the following. The general equation of motion for the overall system is given as follows: 8 9 2 38 9 Fxx 0 0 0 > Fx > x> > > > > > > > = 6 0
where (x, y, z, y) represent the linear and torsional vibrations produced by cutting forces (Fx, Fy, Fz) and torque (Ty), respectively. The direct (Fxx, Fyy, Fzz,Fyy) and cross (Fzy, Fyz) transfer functions of the plunge mill at its tip are expressed by their modal parameters as Fab ðsÞ ¼
H o2nh =kh Da X ¼ , F b h¼1 s2 þ 2zh onh s þ o2nh
(7)
where ab ( a or b is x, y, z, or y) denotes the displacement of the cutter center coordinate a when cutting force F is applied in direction b. H is the total number of modes in the system, h represents each of these modes and onh, kh, and zh are the natural frequency, modal stiffness, and damping ratio, respectively. The axial and torsional vibrations are coupled in this particular cutter, which are represented by crosstransfer functions (Fzy, Fyz). Note that while the plunge mills are always flexible in lateral directions, the contributions of axial, torsional, and coupled torsional–axial modes very much depend on the geometry of cavities between the teeth. The frequency response functions in rigid directions can be set to zero in Eq. (6) which represents a general case. The fourth-order Runge–Kutta method is used for the time domain, numerical integration of the differential Eq. (6) which leads to the prediction of vibrations for the applied cutting forces and torque on the plunge mill. The total displacement of the tool is evaluated by summing the vibrations contributed by all natural modes: DaðtÞ ¼
H X
(8)
Dah ðtÞ,
h¼1
The dynamic chip load distribution along the cutting edge is evaluated by predicting the present and previous cutting edges positions on the chip surface, which requires tracking the cutter center position of the plunge mill (Xc, Yc, Zc, yc), which can be described using the vibration displacement and feed per tooth (ft) as follows: 9 9 8 8 9 8 X c ðtÞ > xðtÞ > > > Rr sin f > > > > > > > > > > > > > > > = = = < Y c ðtÞ > < yðtÞ > < R cos f > r ¼ þ , (9) NcOt Z c ðtÞ > zðtÞ > > > > > > > > > > > 2p > > > > > > > > ; > > : : : yc ðtÞ ; yðtÞ ; Ot |fflfflfflfflfflfflfflfflffl ffl {zfflfflfflfflfflfflfflfflffl ffl } |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl} CUTTER CENTER
RIGID BODY
VIBRATIONS
where Rr represents the radial eccentricity of the cutter center. The cutter center history is stored in an array as a function of cutter rotation angle index (j) or time, which is used for dynamic uncut chip thickness calculation. 3.2. Dynamic uncut chip thickness model
Fig. 3. Time domain simulation procedure for dynamic plunge milling.
Montgomery and Altintas [4] developed an exact kinematic model of dynamic milling by digitizing the surface finish at discrete time intervals. Although their model accurately predicts the exact chip thickness with regenerative vibration effects in milling process, it is
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Fig. 4. Dynamic uncut chip thickness model in plunge milling.
computationally costly and not robust when torsional vibrations are included. An alternative but still an accurate kinematic model of dynamic chip thickness is formulated here by considering the structural vibrations in the lateral (x, y), axial (z), and torsional (y) directions at the cutter center. As shown in Fig. 4, the dynamic chip thickness changes with time by reflecting the regenerative vibration marks left on either side of the chip surface by the present and previous teeth in plunge milling. When the present and previous cutting edge positions are at the same position (Xe ¼ Xem, Ye ¼ Yem) in XY plane, the dynamic chip thickness is defined as the difference between z position (Ze) of the present and z position (Zem) of the previous cutting edge positions. The edge positions ((Xe, Ye, Ze) and (Xem, Yem, Zem)) are calculated using the present and previous cutter center positions ((Xc, Yc, Zc), (Xcm, Ycm, Zcm)) and the cutter geometry information. Since the cutter center history is updated and saved in an array as a function of the cutter rotation angle index (j) or simulation time interval, the dynamic chip load produced by the present and previous teeth can be evaluated easily. The method of tracking the previous cutting edge position is explained in the following. As the angular position (fem) of the previous cutting edge approaches (fe) of the present cutting edge with respect to previous cutter center position (Xcm, Ycm), the previous (Xem, Yem) and the present cutting edge positions (Xe, Ye) become the same. The present and previous cutting edge positions can be evaluated from the time history of the cutter center when fem is equal to fe. The angular position (fm) of the previous cutting edge is located from the algorithm presented in Fig. 5. First, the initial previous cutter rotation angle index (jm) is estimated in order to
locate the previous angular position of the cutter by rotating the cutter backward by a pitch angle: j m ¼ j mj p
ðm ¼ 1; 2; . . .Þ,
(10)
where jp ¼ fp/Dyr and m is the index of the previous tooth passing period. The previous cutter center information (Xcm, Ycm, Zcm, ycm) is extracted according to the index jm. Using the previous center positions, two location angles (fem, fe) of the present and the previous cutting edges are evaluated as follows: fe ¼ tan
ðX e X cm Þ ; ðY e Y cm Þ
fem ¼ iprev fp þ ycm ,
(11)
where iprev is calculated as the remainder of i+m/N, which indicates the tooth index at the previous cutter rotation angle. The lateral and torsional vibrations cause the difference (yv) between two angular positions (fem, fe): yv ¼ fe fem ¼ tan
ðX e X cm Þ fem , ðY e Y cm Þ
(12)
when yv approaches zero, two angles fem and fe as well as two cutting edge positions (Xe, Ye) and (Xem, Yem) match at the present cutting edge element. If yv is not zero, jm is modified using Eq. (13) and the procedure is applied until yv becomes zero: jm ¼ jm þ
yv . Dyr
(13)
The dynamic chip thickness is calculated using Z positions of the present and the previous cutting edges when yv approaches zero. First the present cutting edge
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The dynamic uncut chip thickness can be calculated as follows: hm ði; j; k; mÞ ¼ Z em Z e ,
(16)
The same procedure is applied for calculating the possible dynamic uncut chip thicknesses (hm) which could be generated by the present and the previous teeth at the previous tooth passing periods (m ¼ 1; 2; 3 . . .). The minimum of the possible dynamic uncut chip thicknesses is selected as the real dynamic chip thickness. If the dynamic chip load is less than zero, the tool is assumed to jump out of cut in order to account for the nonlinearity in the process: hði; j; kÞ ¼ max½0; minðhm ði; j; k; mÞÞ.
(17)
The evaluated dynamic uncut chip thickness is applied on the tool tip according to the force expression given in Eq. (5). 4. Experimental validation A series of plunge milling tests have been conducted to validate the dynamic plunge milling model presented in the article. 4.1. Mechanistic identification of cutting force coefficients
Fig. 5. The algorithm to evaluate dynamic chip thickness. (iprev indicates the tooth index at previous cutter rotation angle; jm is previous cutter rotation angle index; hm denotes the possible dynamic chip thicknesses).
The accuracy of predicting the cutting forces and resulting vibrations are highly dependent on the accurate identification of cutting force coefficients. Since the chip load per tooth is small and the insert geometry can be too complex in plunge milling, the cutting coefficients are identified as nonlinear functions of the chip thickness as follows. The intermittent plunge milling tests, (Fig. 1(c)) are used to identify cutting force coefficients. The cutting force equations given in Eq. (5), are re-written in matrix form as 9 8 9 2 38 A11 A12 A13 > > = = < Fx > < Kt > 7 Fy ¼ 6 (18) 4 A21 A22 A23 5 K a , > > > ; ; :F > : A31 A32 A33 Kf z where
location is calculated from the cutter center position as shown below:
A11 ¼
ð14Þ
where Rai is the axial run-out of ith tooth, and Xc, Yc, Zc, and yc are obtained from Eq. (9). The Z position of the previous cutting edge is calculated as follows: Z em ¼ Zcm þ l d tan cr þ Raiprev , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l d ¼ ðX e X cm Þ2 þ ðY e Y cm Þ2 .
ð15Þ
ð cos fÞ hðfÞ Da,
n¼0 k¼0
X e ¼ X c þ rk sinðyc þ ifp Þ; Y e ¼ Y c þ rk cosðyc þ ifp Þ, Z e ¼ Zc þ Rai þ ðrk lÞ tan cr ,
N 1 N k 1 X X
A12 ¼
N 1 N k 1 X X
ðsin fÞhðfÞ Da;
A13 ¼ 0,
n¼0 k¼0
A21 ¼
N 1 N k 1 X X
ðsin fÞhðfÞ Da,
n¼0 k¼0
A22 ¼
N 1 N k 1 X X
ðcos fÞhðfÞ Da;
n¼0 k¼0
A23 ¼ 0,
ARTICLE IN PRESS J.H. Ko, Y. Altintas / International Journal of Machine Tools & Manufacture 47 (2007) 1351–1361 Table 1 Identified cutting coefficients in plunge milling of aluminum alloy Al7050T7451, see Fig. 2 for cutter geometry Kt (kN/mm2)
Ka (kN/mm2)
Kf (kN/mm2)
C1 ¼ 9.887 C2 ¼ 0.796 C3 ¼ 0.358 C4 ¼ 0.769
C1 ¼ 1.662 C2 ¼ 0.652 C3 ¼ 4.186 C4 ¼ 3.375
C1 ¼ 4.095 C2 ¼ 0.721 C3 ¼ 2.723 C4 ¼ 1.577
A31 ¼ 0;
A32 ¼ 0;
A33 ¼
N1 k 1 X NX
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hðfÞ Da.
n¼0 k¼0
The cutting forces are measured during the chatter vibration free plunge milling tests, and geometric parameters (A) are evaluated at each time or angular position of the cutter. The cutting coefficients (Kt, Ka, and Kf) are evaluated by applying the least squares to Eq. (19). The nonlinearity in the cutting coefficients are
Table 2 Modal parameters of the plunge mill attached to Mori Seiki SH403 with HSK 63 interface Mode #
Natural frequency (Hz)
Damping ratio (z)
Modal stiffness (k)
Dynamic stiffness (2 kz)
xx
1 2 3
508 3040 4035
0.0235 0.0242 0.0179
121.00 N/mm 41.10 N/mm 67.50 N/mm
5.68 N/mm 1.99 N/mm 2.42 N/mm
yy
1 2 3
515 3100 3961
0.0510 0.0305 0.0183
44.30 N/mm 80.40 N/mm 51.00 N/mm
4.52 N/mm 4.90 N/mm 1.87 N/mm
zz
1 2
321 405
0.0482 0.0379
242.00 N/mm 790.00 N/mm
23.30 N/mm 59.80 N/mm
zy
1
12090
0.00153
21.20 Nm/mm
0.0649 Nm/mm
yy
1
12000
0.00243
1.106E+04 Nm/rad
53.80 Nm/rad
Fig. 6. Comparison of the predicted and measured cutting forces under chatter-free cutting conditions. Work material: Al7050-T7451, Cutter: see Fig. 2. (a) The predicted and measured cutting forces for two revolutions under fully immersed plunge milling. (a-1) Cutting conditions: spindle speed ¼ 1000 rpm, feed per tooth ¼ 0.075 mm/tooth, radial depth of cut ¼ 2.5 mm. (a-2) Cutting conditions: spindle speed ¼ 1000 rpm, feed per tooth ¼ 0.075 mm/tooth, radial depth of cut ¼ 6.5 mm. (b) The predicted cutting forces and the measured ones for two revolutions under intermittent plunge milling. (b-1) Cutting conditions: spindle speed ¼ 1000 rpm, feed per tooth ¼ 0.05 mm/tooth, radial depth of cut ¼ 4 mm. (b-2) Cutting conditions: spindle speed ¼ 1000 rpm, feed per tooth ¼ 0.125 mm/tooth, radial depth of cut ¼ 3 mm.
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represented by C1 C2 þ C2, (19) 1 þ ðh=C 3 ÞC 4 which indicates that the coefficients have high values at low chip loads, but approaches to a constant value at large chip loads [7]. The identified cutting coefficients for plunge milling of Aluminum alloy is given in Table 1.
K¼
4.2. Plunge milling tests A series of plunge milling tests were conducted at various radial immersion conditions to validate the process model. The structural modal parameters of the plunge milling tool used in the experiments are given in Table 2. While experimental modal tests for low frequencies were performed with Kistler hammer 9722, a PCB 086D80 miniature impulse hammer was used for modes with frequencies higher than 1.5 kHz in order to expand the frequency bandwidth of the impact excitation. Kistler
accelerometer model 8778A00 was attached to the plunge mill to measure the vibrations. The work material was Al7050-T7451 and the cutter geometry is given in Fig. 2. One insert had 20 mm axial and 1.5 mm radial measured run-outs relative to the reference insert, which are included in the simulation model. The machine tool was a Mori Seiki SH403 Horizontal Machining Center, and a Sandvik CG20 HSK63 holder was used during cutting tests. Sample plunge milling test results at n ¼ 1000 rpm spindle speed but at various immersion and feed rates are given in Fig. 6, where the process was free of chatter vibrations. The predicted and measured cutting forces are in good agreement in all cases, which indicates the correctness of rigid body kinematics of plunge milling, chip thickness evaluation from digitized surfaces cut by successive teeth, and nonlinear cutting coefficient model. The figures show that when the cutter is fully immersed, like in drilling (see Fig. 1(b)), the feed force in plunge direction (Fz) is constant, and in plane cutting force (Fx, Fy) amplitudes are small due to cancellation of forces
Fig. 7. Analysis of measured cutting force in case of chatter. Cutting conditions: spindle speed ¼ 16000 rpm, feed per tooth ¼ 0.075 mm/tooth, radial depth of cut ¼ 5 mm. Work material: Al7050-T7451, Cutter: see Fig. 2. (a) Measured cutting forces Fx and Fy for one revolution. (b) Measured cutting force Fz. (c) FFT of the measured Fx. (d) FFT of the measured Fy. (e) FFT of the measured Fz. (f) FFT of the measured sound.
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generated by two symmetric inserts, see Fig. 6(a). The oscillation in lateral forces (Fx, Fy) is due mainly to run-out which upsets the chip load balance between the two inserts. When the radial immersion is intermittent as illustrated in Fig. 1(c), the cutting forces become periodic at tooth passing intervals, see Fig. 6(b), due to loss of force cancellation. Since the accuracy of the simulation model is verified in chatter-free cutting conditions, the proposed dynamic model can be relied on in predicting dynamic milling more comfortably. Fig. 7 shows the measured cutting forces, Fourier spectrum of cutting forces and sound record when the system chattered at the spindle speed of 16,000 rpm with feed rate of 0.075 mm/tooth during a full immersion plunge milling test. Although the axial force Fz should be constant in full immersion cut, it fluctuates at 12360 Hz chatter frequency which is caused by the coupled torsional–axial mode of the plunge mill, see Table 2. The spectrum of the
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sound shows harmonics which are spread on either side of the chatter frequency at tooth passing frequency (533.3 Hz) intervals [12]. The simulation of the same cutting conditions is shown in Fig. 8, which shows the growth of chatter vibrations within ten spindle revolutions. Similar to the measured values, the predicted plunge milling forces clearly indicate the chatter occurring at 12137 Hz. The small discrepancy between the simulated and measured chatter frequency is mainly due to resolution of FFT and numerical integration of the delayed differential equations in evaluating the chip thickness. The high-frequency component is generated by torsional–axial vibration coupling effect, where the axial displacement (zy) is due to the applied torque as shown in Table 2. Unlike cutting forces Fx and Fy, torque Fy generated by each tooth is not canceled and influences the axial displacement by the modal parameter zy. Here, the torsional vibrations are
Fig. 8. Analysis of simulation results in case of chatter. Cutting conditions: same as in Fig. 7. (a) Predicted cutting forces for 10 revolutions. (b) Predicted cutting force for one revolution. (c) Predicted torsion force Ty. (d) Cutter deflection in Z direction. (e) FFT of the predicted Fx. (d) FFT of the predicted Fz.
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Fig. 9. The solid model of the plunge mill, and its axial deformation when torsion is applied to cutting edge in FE model.
translated as axial displacements of the cutter which directly affect the dynamic chip thickness. Finite Element model of the plunge mill is constructed to make sure the dominant mode is due to torsional–axial coupling as shown in Fig. 9, which gives a similar natural frequency (around 12,600 Hz) and axial displacement when torsion is applied. Although the axial modes, the influence of axial mode on torsional displacements, as well as the pure torsional mode are given in Table 2 for completeness, they have very little influence on the chatter stability in this particular set-up. When the speed is increased to 17,142 rpm, the measured and predicted cutting forces are compared in Fig. 10. Stable cutting is found in both simulation and experiment at this cutting condition and the predicted cutting forces are in good agreement with the measured ones. It is verified that stable cutting condition, which leads to high productivity can be pursued using this time domain solution, and that process can also be accurately predicted prior to real machining. 5. Conclusions The time domain model of the dynamic plunge milling process allows prediction of cutting forces, torque, power, and vibrations in plunge milling operations as a function of tool geometry, structural dynamics of the machine and cutting conditions. The cutting forces in plane milling act in lateral directions, hence the major vibrations are dominated by
Fig. 10. Comparison of the measured and predicted cutting forces under stable cutting condition with high spindle speed. Cutting conditions: spindle speed ¼ 17142 rpm, feed per tooth ¼ 0.075 mm/tooth, radial depth of cut ¼ 5 mm. Work material: Al7050-T7451, Cutter: see Fig. 2. (a) The measured cutting forces. (b) The predicted cutting forces.
the bending modes of the tool–holder–spindle assembly. However, the paper shows that the major cutting forces act in the torsional and cutter-spindle axis directions in plunge milling. The lateral forces can be canceled by using symmetric teeth on the plunge mills. While the spindle axis is usually most rigid, the chip evacuation cavities weaken the cross-section of the plunge mills which lead to torsional flexibility in plunge mills. The torsional vibrations are transmitted as axial displacements through structural coupling, and affect the chip thickness regeneration most, hence causing chatter. The torsional chatter can be reduced by designing the plunge mills with strengthened chip evacuation cavities. The lateral vibrations can be reduced by minimizing the tool run-outs and using symmetric tools.
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Acknowledgments This work is supported by the Post-doctoral Fellowship Program of Korea Research Foundation (KRF) for the lead author, and by the National Sciences and Engineering Research Council of Canada (NSERC) and Pratt and Whitney Canada under Industrial Research Chair and Strategic Grants of the second author. The cutting tools and tool holders are provided by Mitsubishi Materials and Sandvik Coromant Companies, and the high speed Mori Seiki SH403 horizontal machining center is contributed by Machine Tool technology Research Foundation (MTTRF) Company. References [1] S. Wakaoka, Y. Yamane, K. Sekiya, N. Narutaki, High-speed and high-accuracy plunge cutting for vertical walls, J. Mater. Process. Technol. 127 (2002) 246–250. [2] Y. Li, S.Y. Liang, R.C. Petrof, B.B. Seth, Force modelling for cylindrical plunge cutting, Int. J. Adv. Manuf. Technol. 16 (2004) 863–870. [3] J. Tlusty, F. Ismail, Basic nonlinearity in machining chatter, CIRP Ann. 30 (1981) 21–25.
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