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Time domain prediction of milling stability according to cross edge radiuses and flank edge profiles
Author’s Accepted Manuscript Time domain Prediction of milling stability according to cross edge radiuses and flank edge profiles Jeong Hoon Ko www.elsevier.com/locate/ijmactool
PII: DOI: Reference:
S0890-6955(14)40016-6 http://dx.doi.org/10.1016/j.ijmachtools.2014.11.004 MTM3010
To appear in: International Journal of Machine Tools and Manufacture Received date: 30 August 2014 Revised date: 11 November 2014 Accepted date: 11 November 2014 Cite this article as: Jeong Hoon Ko, Time domain Prediction of milling stability according to cross edge radiuses and flank edge profiles, International Journal of Machine Tools and Manufacture, http://dx.doi.org/10.1016/j.ijmachtools.2014.11.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Time Domain Prediction of Milling Stability according to Cross Edge Radiuses and Flank Edge Profiles Jeong Hoon Ko* Singapore Institute of Manufacturing Technology, 71 Nanynag drive, 638075 Singapore * Corresponding author. Tel.: +65-6793-8592; fax: +65-6791-6377. E-mail. address:[email protected]
Abstract This article proposes a time domain model for predicting an end milling stability considering process damping caused by a variety of cross edge radiuses and flank profiles. The time domain model of calculating indentation areas, as well as regenerative dynamic uncut chips, is formulated for the prediction of the stabilizing effect induced by interference areas between the edge profiles and undulation left on a workpiece. The interference area generates forces against the vibration motion, which acts as a damping effect. In the model, the present and previous angular position of cross radiuses and flank edge profiles are located to calculate the dynamic uncut chip as well as indentation area based on a time history of the dynamic cutter center position. The phenomenon that chatter is damped according to cross edge radiuses and flank edge profiles is successfully simulated with the proposed dynamic model and validated through the extensive experimental tests. Keywords: time domain chatter model, cross-flank edge profiles, end milling stability, process damping
Nomenclature
hl
h f
local helix angle radial rake angle of cutting edge cutting edge angle helix angle flank elemental edge angle
Lf
flank elemental length
p
pitch angle
Fn (i, j , k ) normal pressure force Ff (i, j , k ) frictional force
n b Tc Nf
a Na
e
T ( ) c
unit vector tangent to the cutter edge unit vector normal to rake face unit vector on the rake surface and perpendicular to the cutter edge chip flow vector the number of flutes length of edge element along the radial direction. total disk number rotation angle of the cutter cutting edge location angle transformation matrix flute spacing angle
hc
uncut chip thickness
hcr
rescaled uncut chip thickness
ds Kn Kf θc Rr R er era rk
incremental cutting edge length normal cutting force coefficients frictional cutting force coefficients chip flow angle radial runout cutter radius cross edge radius separation angle cutting edge position from cutter center
1. Introduction Machining chatter often turns a great deterrent against high productivity and machining quality. There have been extensive research works on the mechanics and dynamics of milling processes. Tobias and Tlusty introduced first chatter stability laws in the frequency domain [1,2]. Sridhar presented the time domain solution with two coupled, delayed differential equations with time-varying coefficients [3]. Minis and Yanushevsky proposed the first analytical solution of the milling stability using Floquet's theory by advancing from Sridhar's formulation [4]. Altintas and Budak developed a general and closed solution of the milling stability in the frequency domain [5,6]. Some articles proposed added stability lobes to simulate the low radial immersion cutting during high speed machining [7,8]. Olgac et. al. included both single and multiple time delays and validated their stability law on uniform and variable pitch cutter results [9]. Ko and Altintas proposed mechanistic and dynamic models in time and frequency domain for plunge milling processes [10,11]. The developed stability model can predict a torsional-axial vibration as well as lateral vibrations. Eksioglu and Altintas [12] proposed a general formulation of flexible mills and workpiece at high axial and small radial depths of cuts. Recently, modeling of milling process damping has been improved by a few researchers [13-16]. Tunc and Budak [15, 16] proposed an indentation model based on the existence of a separation point on the cutting edge in order to estimate process stability. Eyniyan and Altintas [17, 18] proposed a dynamic cutting force model and a novel experimental technique for the identification of dynamic coefficients with a piezo actuator set-up. Researchers mentioned that the interaction between a wavy surface and a cutter flank should be further investigated as it affects the process damping. Regarding the edge radius effect, Waldorf et al. [19] proposed two ploughing models for static / orthogonal model and Endres [20] worked on the force balance under the deformation zone for static force prediction. Albrecht [21] investigated that the existence of hone radius causes the workpiece material to be extruded against the cutting tool, which increases process damping. Wu [22] investigated the relation of the penetration depth to the uncut chip thickness, vibration amplitude and the instantaneous shear angle. Shawky and Elbestawi [23] estimated the depth of penetration due to cross edge radius and Ranganath et al. [24] estimated the separation angle in order to incorporate indentation forces into the overall force model. Depending on tool wear or tool edge preparation, the profiles of the cross edge and its adjacent flank edge vary which affects the surface quality as well as stability significantly. Fig. 1 shows the examples of different cross edge radiuses according to insert or solid type of end mills. So, it is important to consider the cutting edge radiuses of the 2
tool as one of the variables for predicting machining stability. In addition, the profile of flank edge adjacent to cross edge may change according to tool wear, which need to be measured and modeled to simulate its damping effect accurately.
Fig. 1 cross edge measurement according to the type of cutting edge
This paper proposes a fast and comprehensive time domain model which simulates a dynamic uncut chip as well as the interaction of cross-flank edge profiles with the undulation of machined surface. It can predict how the edge profiles affect the milling stability by tracking the interaction between cross-flank edge profiles and undulation. Contact forces between cross–flank edges and wave on the machined surface contribute to the dynamics of the cutting process by increasing the overall damping. The elemental contact forces are integrated by simulating time-dependent elemental indentation volume between cross-flank edge and undulation profiles, which act against the cutting forces of removing the dynamic uncut chip. By tracking dynamic cutter center positions determined by vibration, runout, and feed motions, the time-dependent angular positions of the edge profiles as well as the undulation can be rapidly and precisely located. The angular positions are used to simulate the dynamic uncut chip as well as the contact area of cross-flank edge profiles against the undulation left on the machined surface. According to a variety of edge radiuses and flank profiles, the milling stabilities are experimentally compared and analyzed. The corresponding simulation results are presented and validated with the experimental ones on the damping effect with cross edge radiuses and flank edge profiles.
2. Time domain milling model considering cross-flank edge profiles Figure 2 illustrates the proposed time domain solution process. Firstly, after the cross edge radius and flank edge profiles are measured, tool geometry information, modal parameters, cutting conditions, and workpiece info are input to the simulation model. The edge profiles are digitized to simulate elemental forces and dynamic uncut chip thickness. After solving the dynamic equation using a numerical time integral scheme, the cutter center positions at 3
present and previous rotations are updated. By extracting the time history of the cutter center positions, the angular positions of the edge elements and the undulation on the surface can be estimated to formulate the dynamic uncut chip thickness as well as the intersection area between the edge profiles and the wave on the machined surface. The estimated dynamic uncut chip thickness and contact area are input to cutting and contact force model. Finally, the time-dependent cutting forces and vibrations can be continuously simulated using this procedure in order to evaluate the milling stability.
Fig. 2 Simulation procedure of time domain solution
The nose region of a cutting tool has a finite cross edge radius depending on tool edge preparation or wear. In front of the rounded portion of the tool nose, there is a separation angle (era ) which distinguishes the indentation from the shearing. The profile of flank edges may have curvature or different flank angles, which can be measured using optical devices and digitized into segmental flank angles. Fig. 3 shows the elemental cutting forces as well as indentation ones, which are integrated into total forces.
4
Fig. 3 unit vectors and force components component
The chip load and corresponding differential loads for each edge element are evaluated and digitally integrated to predict the total forces in three directions. The cutting forces acting on the rake surface of the disk element are divided into two orthogonal components: the normal pressure force, dFn (i, j , k ) , and the frictional force,
dF f (i, j , k ) . It can be obtained from
dFn (i, j , k ) K n ndAc (1)
dFf (i, j, k ) K f KnT c dAc where dAc h( ) cos r (a / cos hl ) , and n and T c are the unit vectors shown in Fig. 3, respectively [26].
cos cos hl T ( , hl ) sin cos hl sin hl
sin cos 0
cos sin hl sin sin hl cos hl
Using the transformation matrix considering local helix angle ( hl ) and each edge element’s angular location ( ),
dFn and dF f can be decomposed into three orthogonal force components in Cartesian coordinates as follows: dFx (i, j,k) [ K n (cos r cos cos hl sin r sin )
(2)
f 1 1 cos cos hl cos r sin 1 cos r cos sin hl ) f2 f2 2 r cos ) [ K n (cos r sin cos hl sinf K n K f cos c (sin r
dFy (i,j,k)
1 1 f1 1 f1 K cos cos cos r sin sin hl ) K f sin c (r f sin sin hl cos sin hl )] dA n KKf ncos c (sin r cos c f f 2f 2 f 2 2
K n K f sin c (
2
f1 1 cos sin sin hl )]dAc f2 f2
dFz (i, j,k) [ K n sin hl cos r K n K f cos c ( sin r K n K f sin c
1 cos hl ]dAc f2
f 1 sin hl 1 cos r cos hl ) f2 f2 5
The relationship between the rescaled uncut chip thickness hcr(i, j, k) and ln(Kn) can be readily derived using the Weibull function.
ln( Kn ) A1 ( A1 A2 )e( A3hcr )
A4
(3)
Kf and c can be represented as a nonlinear function dependent only on instantaneous uncut chip thickness hcr which is calculated using Eq. (4). K f B1 ( B1 B2 )e ( B3hcr )
c
B4
(4)
C1 C2 C2 1 (hcr / C3 )C4
The cutting force coefficients ( Kn, , Kf , and c ) can be identified using a mechanistic approach using a few cutting force measurements[26]. For example, the coefficients of Al 6061-T6 are calculated as A1 = 7.054, A2 = 10.303, A3 = 1.218, A4 = 1.040, B1 = 0.613, B2 = 0.912, B3 = 0.391, B4 = 2.772, C1 = 0.560, C 2 = 0.434, C3 = 4.548, and C 4 = 0.231. For S304, the coefficients of K n are defined as A1 = 7.467, A2 = 10.303, A3 = 1.218, A4 = 1.040. For estimating elemental contact forces which act as elemental process damping forces, the mean friction coefficient ( ) on the contact surface is assumed to be constant. And the normal force ( dPn ) and frictional force components ( dPf ) can be defined as follows:
dPn (i, j , k ) K d dV
(5)
dPf (i, j, k ) dPn where K d is the specific ploughing force over the unit volume dV of displaced work material, is the mean friction coefficient on the contact surface. K d and can be estimated using an iterative method based on the proposed time domain model compared to other approaches[15, 22, 25]. Considering the angular position ( ef ) of cross edge and flank profiles, the elemental indentation forces are transformed into Cartesian coordinates. The indentation is attributed to the actual contact between the workpiece and the tool nose region, which includes the tool cutting edge and its adjacent flank face. The indentation load and corresponding differential loads for each edge element are evaluated and digitally integrated to predict the damping forces acting against vibrations.
dFpx (i, j , k , f ) dPx ' cos(ef ) dPy ' sin(ef ) dFpy (i, j , k , f ) dPf sin(ef ) dPn cos(ef ) (6)
6
Here dPx ' and dPy ' for flank edges are defined as follows:
dPx ' dPff cos f dPnf sin f
(7)
dPy ' dPff sin f dPnf cos f For cross edges, the force components are defined as follows:
dPx ' dPfc cos c dPnc sin c
(8)
dPy ' dPfc sin c dPnc cos c
In the elemental indentation force components, c changes according to the type of tooling as well as tool wear. Since the flank edge may have curvature, the flank angle f may have different values according to elemental flank edge index f .
Finally, total forces can be defined as follows
F(j) dF (i, j, k ) dFp (i, j, k , f ) i
k
i
k
f
(9)
The dynamic equation for milling system can be formulated as follows:
x xx y yx z zx
xy yy zy
xz Fx yz Fy zz Fz
(10)
where M is modal mass, C is modal damping, K is modal stiffness. Coordinate ab ( a or b is x, y, z or ) denotes the displacement of the cutter center coordinate a when cutting force F is applied in direction b. From experimental modal analysis, the direct transfer function of the structure along the x, y, z direction at the tip of the end mill is obtained in the Laplace domain as,
ab (s)
wnh 2 / kh a Nh 2 Fb h1 s 2 h wnh s w2nh
(11)
where Nh is the total number of modes in the system, h represents each of these modes and wnh , k h , and h are the natural frequency, modal stiffness, and damping ratio, respectively. The fourth order Runge-Kutta equation is used for the numerical integration of the differential Eq. (10). The total displacement of the tool is evaluated by summing the vibrations contributed by all natural modes. Nh
a (t ) a h ( t )
(12)
h 1
7
Finally, the center position of the end mill ( X c , Yc , Z c ) can be described using the vibration displacement and feed per tooth ( f t ) as follows:
X c (t ) feed x t Rr sin x(t ) Yc (t ) feed y t Rr cos y (t ) Z (t ) z (t ) c feed z t Cutter Center
Rigid Body Motion
(13)
Vibration
where Rr represents the radial runout of the cutter center.
3. Estimation of dynamic uncut chip and indentation area This section explains the fast and accurate simulation approach on dynamic uncut chip thickness as well as flankwave interaction volume. The proposed simulation method takes less than 1.5 Sec for predicting the process of one cutting condition with a simulation time interval digitized at least ten times smaller than the period of the highest frequency among modal frequencies.
3.1 Previous and present cutting edge positions on vibration wave Firstly, the cutting edges including cross and flank edges are divided into a finite number of small differential elements in the cutting edge curve. The angular position e (i, j , k ) of the kth edge element of the tooth i at the jth angular position of the cutter is described by
e (i, j, k ) r ( j ) (i 1) p
k da tan h , r ( j ) j r R
(14)
where r is the rotation angle of the cutter measured from y axis, and p 2 / N is the pitch angle. In the proposed model, the present and previous edge positions can be calculated using the present and previous cutter center positions and cutter edge geometry information. Using Eq. (10), the cutter center position ( X c , Yc , Z c ) is updated and saved according to incremental angular value r corresponding to the simulation time interval considering the period of highest natural frequency. Since the cutter center positions are saved according to cutter rotation angle , the previous cutter center positions ( X cm , Ycm , Z cm ) can be extracted by locating the previous cutter rotation angle m . Fig. 4 summarizes the overall algorithm of dynamic uncut chip thickness calculation method by using previous and present cutter center positions.
8
(a) Angular mismatch
v
(b)
e
has the same value as
em
(c) Algorithm of tracking present and previous cutting edge for dynamic uncut chip thickness
Fig. 4 Dynamic uncut chip thickness calculation algorithm
9
In order to consider the phase of the vibration wave, the dynamic uncut chip thickness is defined as the distance between the present and previous cutting edge locations which are aligned to the radial direction of the present carter center. To calculate the radial directional dynamic uncut chip thickness, the previous cutting edge’s angular location (
em ) is to match with the present cutting edge’s angular location ( e ). In the case that e has the different value as em over the tolerance (ε) as shown in Fig. 4(a), v ( e - em ) should be added to previous cutter center location (
cm ) in order to relocate previous cutting edge. Finally, when e has the same value as em within the tolerance as in Fig. 4(b) after a few iterations, two cutting edges’ distance is calculated as dynamic uncut chip thickness. The algorithm incorporates the time delay of the vibration phase by accurately locating two cutting edges forming vibration wave as explained in Fig. 4(c). The coordinates of the points along the cutting edge are evaluated by using the cutter geometry information. The coordinates ( Xe, Ye , Ze ) of a point along the cutting edge of tooth j are expressed as:
X e, j (t ) X c (t ) rk sin( j ), Ye, j Yc (t ) rk cos ( j ), Ze,j (t ) Z c Raj (rk l ) tan r Where
(15)
rk ( X e X c )2 (Ye Yc )2 j t ( j 1) p
So the possible dynamic uncut chip thickness is evaluated by subtracting the radial distance of the present teeth from the previous one with reference to the cutter center as follows:
hkm ( j (t )) ( X e X c )2 (Ye Yc )2 ( X em X c )2 (Yem Yc )2
(16)
The final dynamic uncut chip thickness is selected as the minimum value among hkm and should be larger than zero as follows:
h(i, j , k ) Max[0, Min( hkm (i, j, k , m))]
(17)
where hkm(i, j, k, m) represents the possible uncut chip thicknesses. m is previously passed teeth (m = 1 , …, Nf) at the same edge location angle as the present angle and h indicates the real uncut chip thickness.
3.2 Interaction of cross-flank edges against undulation of the machined surface In order to consider the indentation volume between cross-flank edge profile and the undulation left on the surface, the cross and flank edges are divided into a finite number of small differential elements. In Fig. 5(a), the angular position, ef of the lth cross-flank element of the kth disk of the tooth i at the jth angular index is defined by
ef (i, j, k , f ) r ( j ) (i 1) p
k da tan h f (l ) R
(18)
where f (l ) , the angular position of the cross-flank edge element is estimated as follows.
10
f (l ) cos 1 (
dey (l ) dex 2 (l ) dey 2 (l )
)
(19)
Where dex (l ) l df , dey (l ) R er (er 2 (l df )2 ) for the section of cross edge radius er and l = 0.. n1. The
position
of
the
elemental
flank
edges
is
defined
as
dex (l ) (l n1 ) df cos( f )
,
dey (l ) (l n1 ) df sin( f ) where l = n1+1…n2. Here, n2 is the total digitization number of cross and flank edges. As in Fig. 5(b), flank edges may have curvature or may have different angles, which are divided into smaller segment with various angles ( f 1 , f 2 , f 3 , .. and so on ) according to the profile. The angles can be measured and identified from 3D optical devices. So, depending on the profile of the cross and flank edges, dex and de y can be defined accordingly. The initial previous cutting edge location is estimated as follows.
elm (i, j, k ) mN r (i 1) p , r ( j ) j r , N 360 /
(20)
Then a few iterations are required until its difference ( v ) between ef and elm reduces less than the tolerance ( ) according to the algorithm explained in Fig. 5(c). Finally, the intersection height is calculated as follows
dI h max[0, min( Pef Pelm )] where Pelm ( X elm X c )2 (Yelm Yc )2 and
Pef ( X ef X c )2 (Yef Yc )2
(21) Depending on the undulation of the workpiece and the profile of cross and flank edges, the indentation height has zero or multiple potential values. Its minimum value over zero is the actual indentation height. After this indentation height is estimated, the elemental indentation volume ( dV dI h da df ) are input to the Eq. (5) to calculate the process damping forces. The formation enables the prediction of the stabilizing effect induced by the indentation areas between cross-flank edges and undulation left on machined surfaces according to cross edge radiuses and flank edge profiles.
11
(5-b) cross edge and its adjacent flank (5-a) Calculation of elemental indentation areas on cross and flank edges
angles
(5-c) Algorithm for calculating indentation areas between flank edge components and undulation on the surfaces
Fig. 5 Estimation of angular position of the cross edge and adjacent flank edge elements
4. Comparison of experimental and predicted milling stabilities 4.1 Overall stability changes according to cross-flank edge profiles The profiles of cross-flank edge profiles were generated by gradually wearing out sharp cross edges through 12
repetitive machining. 3D non-contact metrology system was used to measure the accurate values of the edge radiuses as well as flank elemental angles. The values were averaged from a series of measurements of cross edges and digitized into the simulation model. In this paper, three different CERs and flank profiles were generated and categorized as fresh tool, worn tool-1 and worn tool-2 as shown in Fig. 6. With the three sets of the edge profiles as described in Table 1, the milling stability tests were performed and compared over various RPMs and cutting depths for the work materials such as Al6061-T6 and SUS304 as listed in Table 1. Around more than 80 tests were performed so that the predicted milling stabilities can be compared with measured ones according to the different edge profiles.
13
(6-a) Fresh Tool
(6-b) Worn Tool-1
(6-c) Worn Tool-2 Fig. 6. Changes of cross edge radiuses according to tool wear
14
Machine
Roeders milling machine Dia. = 10 mm, No. of flute = 2, Rake angle = 10°, Helix angle = 30°, flank angle = 10° CER = 8.168 µm, f 1 =7.391° L f 1 =100.37 µm Fresh Tool CER = 23.349 µm, f 1 =-24.940° L f 1 =32.247
Milling Cutters with different CER (Cross Edge Radius)
µm, f 2 = 6.5863°, L f 2 =63.763 µm, f 3 = Worn Tool-1
7.2821°, L f 3 =50.537 µm CER = 32.690 µm, f 1 =-11.467° L f 1 =31.115 µm, f 2 = 1.583°, L f 2 =21.214 µm, f 3 =
Worn Tool-2 Cutting Conditions
4.076°, L f 3 =56.617 µm
Feed / tooth = 0.05, AE = 3 mm, AP = 2 mm to 4 mm RPM range from 5000 to 11500 RPMs
Materials Tested AL6061-T6 and SUS 304 Table 1. Experimental milling conditions ( The runout of the tools: 0.004 mm ) For the side milling processes, the modal parameters are experimentally taken for x and y directions as shown in Table 2. The listed natural frequency, damping and stiffness composes the transfer function ab of the tooling on the spindle as defined in Eq. (11). The identified transfer function ab is input to the model as indicated in Eq. (10) in order to simulate the dynamic movement of the end mill. As seen in modal parameters, it is expected that a chatter frequency would be around 4.8 KHz since the dynamic stiffness of the mode no. 1 is least compared to other modes listed in Table 2. Table 2. Measured modal parameters of the end mill attached to Roeders 760 Mode no. xx
yy
Natural Frequency (
Damping Ratio ( )
Hz)
Modal Stiffness (k ) (N/m)
1
4800
0.014
15659818
2
1062.5
0.012
64108769
3
4362.5
0.013
77923634
4
2068.75
0.038
35180967
1
4812.5
0.014
18061694
2
1068.75
0.012
54989075
3
4368.75
0.010
94097720
4
1931.25
0.044
37353541
Figs. 7 and 8 illustrate the changes of overall stability lobes over tested conditions with the comparisons of measured and predicted results. The effects of cutting conditions on the milling stability are clearly compared 15
according to edge profiles of each tool. The experimental machining stability is validated through machined surface quality and sound signal’s FFT. For example, the tested conditions are categorized as “stable” conditions if it dampens the chatter mode with 4.8 KHz in FFT of sound signals and produces surface roughness less than Ra 0.5 µm without chatter mark. As shown in Figs. 7 and 8, the milling stabilities vary according to different tools, which means that cross-flank edge profiles play a critical role in affecting process stability. Since worn tool-2 has a higher CER than other tools, it is understood that as CER increases, the milling process becomes stable due to increased process damping.
(a) Stabilities with fresh tool
(b) Stabilities with worn tool-1
(c) Stabilities with worn tool-2 Figure 7 Changes of stabilities in experimental and simulation results according to cross-edge profiles for milling Al6061-T6 16
(a) Stabilities with fresh tool
(b) Stabilities with worn tool-1
(c) Stabilities with worn tool-2 Figure 8. Changes of stabilities in experimental and simulation results according to cross-edge profiles for milling SUS 304
17
4.2 Comparison and analysis of measured and predicted stabilities according to cross-flank edge profiles In Fig. 9(a), clear chatter marks are shown in the photo of the machined surface with fresh tool while Fig. 9(b) illustrates clean surface with worn tool 2. Accordingly, the sound signal’s FFT illustrates that the chatter frequency around 4.8 KHz appears with the fresh tool as shown in Fig. 9(c) while it is suppressed with the damping effect of the worn tool-2 as in Fig. 9(d).
(9-a) Ra = 2.501µm for new tool
(9-b) Ra = 0.281 µm for worn tool-2
(9-c) Sound signal’s FFT for new tool
(9-d) Sound signal’s FFT for worn tool-2 Figure 9. Comparison of measured surface roughness and sound signals for milling SUS 304 with fresh tool and worn tool-2: RPM 8000, AE 3 mm, AP 2 mm, Feed rate 800mm/min
By inputting the geometry and cutting condition into the proposed time domain model, the dynamic uncut chip is calculated and cutting forces are predicted considering the cutting force coefficients of test materials. For the flankwave contact force coefficients, K d = 22000 N/mm3 and = 0.3 are used for AL6061-T6. K d and are defined as 56000 N/mm3 and 0.3 respectively for SUS304. A separation angle (era ) is identified as 17° for the simulation. As shown in Fig. 10(a), the simulated forces are showing chatter phenomenon for end milling with the fresh tool for the given condition at RPM 8000. However, the oscillating forces with chatter frequency 4.8 KHz are damped out in end milling with worn tool 2 as in Fig. 10 (b). (Please note that the dynamometer used in the experiment has an integrated filter 200 Hz, which is not able to provide measured forces comparable to predicted ones.) In addition, the cutter 18
center position’s movement is simulated at RPM 8000, and FFTs of the simulated displacements are analyzed. The chatter vibration with worn tool-2 is damped as illustrated in Fig. 10(d) while it appears at FFT of displacements for end milling with the fresh tool in Fig. 10(c). So the simulation results have good agreements with the experimental ones in Fig. 9 in terms of the milling stability according to edge profiles.
(10-a) Simulated forces for fresh tool
(10-c) FFT of simulated displacements for fresh tool
(10-b) Simulated forces for worn tool 2
(10-d) FFT of simulated displacements for worn tool-2
Fig. 10 Simulated forces and FFTs of cutter displacements for milling SUS 304 with fresh tool and worn tool-2 for condition RPM 8000, AE 3 mm, AP 2 mm, Feed rate 800mm/min
For the end milling of Al 6061-T6 at RPM 11500, the measured surfaces and FFT of sound signals are compared according to the fresh tool and the worn tool-2 as shown in Fig. 11. It is clear that worn tool-2 causes higher process damping than the fresh tool as the chatter frequency is damped, and the surface roughness is better with worn tool-2.
19
(11-a) Ra = 0.470 µm for fresh tool
(11-c) Sound signal’s FFT for fresh tool
(11-b) Ra = 0.196 µm for worn tool-2
(11-d) Sound signal’s FFT for worn tool-2
Figure 11. Comparison of measured surface roughness and sound signals for milling Al6061-T6 with fresh tool and worn tool-2: RPM 11500, AE 3 mm, AP 2 mm, Feed rate 1150 mm/min According to the time domain simulation for the same cutting condition as in Fig. 11, the simulated cutting force profiles are compared with different tools as illustrated in Figs 12(a) and (b). And the FFT of the displacements of the simulated cutter centers shows different stabilities according to fresh and worn tool-2 in the Figs 12(c) and (d), which match with the experimental stabilities as illustrated in Fig. 11.
20
(12-a) Simulated forces for fresh tool
(12-c) FFT of simulated displacements for fresh tool
(12-b) Simulated forces for worn tool-2
(12-d) FFT of simulated displacements for worn tool-2
Figure 12. Comparison of predicted forces and FFTs of cutter displacements for milling Al6061-T6 with fresh tool and worn tool-2: RPM 11500, AE 3 mm, AP 2 mm, Feed rate 1150 mm/min
Fig. 13 illustrates the change of FFT of measured sound signals according to three tools at RPMs 8000 and 11500. It is observed that RPM 8000 incurs more process damping than RPM 11500 considering that, in end milling with worn tool-1, the chatter frequency around 4.8 KHz is damped at RPM 8000 while the chatter frequency still appears at RPM 11500. In the case of the fresh tool, the chatter frequencies appear at both RPMs 8000 and 11500 while the frequencies are damped out at these RPMs with worn tool-2. The same phenomenon appears in predicted results as well in Fig. 14 which shows FFT of cutter center’s displacements according to RPMs 8000 and 11500. Especially, at RPM 8000, the chatter frequency around 4.8 kHz is damped with worn tool-1 but the frequency still appears at RPM 11500. So it is well validated that the time domain model can predict the effect of cross-flank edge profiles on process damping at different RPMs.
21
fresh tool (chatter)
worn tool-1 (stable)
worn tool-2 (stable)
(a) Stability change according to different edge profiles at 8000 RPM based on FFTs of sound signals
fresh tool (chatter)
worn tool-1 (chatter)
worn tool-2 (stable)
(b) Stability change according to different edge profiles at 11500 RPM based on FFTs of sound signals Figure 13. Comparison of FFT of the measrued sound signals for milling SUS 304 : RPMs 8000 and 11500, AE 3 mm, AP 2 mm, feed per tooth 0.05 mm/tooth
fresh tool (chatter)
worn tool-1(stable )
worn tool-2 (stable)
(a) Stability change according to different edge profiles at 8000 RPM based on FFT of predicted displacements
22
fresh tool (chatter)
worn tool-1(chatter)
worn tool-2(stable)
(b) Stability change according to different edge profiles at 11500 RPM based on FFT of predicted displacements Figure 14. Comparison of FFT of simulated cutter displacements for milling SUS 304 for condition RPMs 8000 and 11500, AE 3 mm, AP 2 mm, feed per tooth 0.05 mm/tooth
5. Conclusion This article proposes the time domain model for predicting the end milling stability according to cross-flank edge profiles over various cutting conditions. The model simulates the interaction areas between cross-flank edge profiles and undulation ones of the machined workpiece, which produces a damping against regenerative vibrations. The time-dependent angular positions of the edge profiles can be accurately located according to the undulation profiles left on workpieces using present and previous dynamic cutter center position determined by vibration and rigid body motion. By dynamically tracking the cutter movements, it predicts overall dynamic forces and vibration generated by the edge profiles against the wavy machined surface. The phenomena that chatter is damped according to the various cross edge radiuses and adjacent flank profiles were successfully simulated with the proposed model and experimentally validated through the extensive experimental tests. The simulation and experimental results of the milling stabilities with the fresh tool, worn tool-1 and worn tool-2 were compared over various cutting conditions for the end milling of Al6061-T6 and SUS304. The chatter frequency 4.8 kHz still appeared at end milling with fresh tool at the tested RPMs, but it was damped with worn tool-1 at spindle RPMs less than 8000 RPM. If RPM increased to 11500 RPM, the frequency appeared even with worn tool-1. However, the end milling with worn tool-2 has a strong process damping effect, which damps the chatter frequency even at 11500 RPM. So, for the variety of the cross edge radiuses and adjacent flank profiles, the machining stability can be accurately predicted using the proposed model. It is expected that the simulation approach will be used to design cross-flank edge geometry for the stabilization of the milling chatter, which will be further investigated in the near future. The research will be useful for a tooling industry regarding the design of the tool geometry. In terms of the tool design, the cross-flank edge profiles have a critical role in terms of the stability as well as the surface quality. Depending on the surface requirement as well as operating parameters, the cross-flank edge profile needs to be optimized so that the machining stability may be secured for the required machined surface quality.
23
Acknowledgements The author appreciates support from Mr. Ng Thai Ee and Mr. Ruben S/O Sukumar from SIMTech. References [1] S.A. Tobias SA, W. Fiswick, Theory of Regenerative Machine Tool Chatter, Engineering, London (1958) 258. [2] J. Tlusty, M. Polacek, Examples of the Handling of the Self-excited Vibration of Machine Tools, FoKoMa, Hanser publisher, Munich (1957) [3] R. Sridhar, R.E. Hohnand, G.W. Long, General Formulation of the Milling Process Equation, ASME Journal of Engineering for Industry 90 (1968) 317-324. [4] I. Minis, T. Yanushevsky, A New Theoretical Approach for the Prediction of Machine Tool Chatter in Milling, ASME Journal of Engineering for Industry 115 (1993) 1-8. [5] E. Budak, Y. Altintas, Analytical Prediction of Chatter Stability in Milling-Part I: General Formulation, ASME Journal of Dynamic Systems, Measurement, and Control 120 (1) (1998) 22-30. [6] E. Budak, Y. Altintas, Analytical Prediction of Chatter Stability in Milling- Part II: Application of the General Formulation to Common Milling Systems, ASME Journal of Dynamic Systems, Measurement, and Control 120 (1) (1998) 31-36. [7] T. Insberger T, G. Stepan, Stability of the Milling Process, Periodica Polytechnica- Mechanical Engineering 44 (2000) 47-57. [8] M.A. Davies,
J.R. Pratt,
B.S. Dutterer, T.J. Burns, The Stability of Low Radial Immersion Milling,
CIRPAnnals— ManufacturingTechnology 49 (2000) 37-40. [9] N. Olgac, R. Sipahi, Chatter Stability Mapping for Simultaneous Machining, ASME Journal of Manufacturing Science and Engineering 127 (2005) 791-800. [10] J.H. Ko, Y. Altitntas, Dynamics and Stability of Plunge Milling Operation, ASME Journal of Manufacturing Science and Engineering 129 (2007) 32-40. [11] J.H. Ko, Y. Altitntas, Time Domain Model of Plunge Milling Operation, International Journal of Machine Tools and Manufacture 47 (2007) 1351-1361. [12] C. Eksioglu, Z.M. Kilic, Y. Altintas, 2012, Discrete-time Prediction of Chatter Stability, Cutting forces, and Surface Location Errors in Flexible Milling Systems, Trans. ASME Journal of Manufacturing Science and Engineering 134 (2012) 1-13. [13] V. Sellmeier, B. Denkena, High Speed Process Damping in Milling, CIRP Journal of Manufacturing Science and Technology 5 (2012) 8-19. [14] T.T. Christopher, L.S. Tony, Analytical Process Damping Stability Prediction, Journal of Manufacturing Process 15 (2013) 69-76. [15] LL.T. Tunc, E. Budak, Effect of Cutting Conditions and Tool Geometry on Process Damping in Machining, International Journal of Machine Tools and Manufacture 2012; 57: p. 10-19.
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[16] L.T. Tunc, E. Budak, Identification and Modelling of Process Damping in Milling, ASME Journal of Manufacturing Science and Engineering135(2) ( 2013) [17] Y. Altintas, M. Eynian, H. Onozuka , Identification of dynamic cutting force coefficients and chatter stability with process damping , Cirp Annals-manufacturing Technology 57(1) 2008 371-374. [18] M. Eyniyan, Y. Altintas, Analytical Chatter Stability of Milling with Rotating Cutter Dynamics at Process Damping Speeds, ASME Journal of Manufacturing Science and Engineering 132(2) (2010) 1-14. [19] D.J. Waldorf, R.E. Devor, S.G. Kapoor, An Evaluation of Ploughing Models for Orthogonal Machining,” ASME Journal of Manufacturing Science and Engineering 121 (1999) 550–558. [20] J. Manjunathaiah, W.J. Endres A New Model and Analysis of Orthogonal Machining With an Edge-Radiused Tool ASME Journal of Manufacturing Science and Engineering 122 (2000) 384–390. [21] P. Albrecht New Development in the Theory of Metal-Cutting Process, Part I. The Ploughing Forces in Metal Cutting ASME Journal of Engineering for Industry 82 (1960) 348–358. [22] D.W. Wu Application of a Comprehensive Dynamic Cutting Force Model to Orthogonal Wave-Generating Processes International Journal of Mechanical Sciences 30(8) (1988) 581–600. [23] A.M. Shawky, M.A. Elbestawi, An Enhanced Dynamic Model in Turning Including the Effect of Ploughing Forces ASME Journal of Manufacturing Science and Engineering 119(1) (1997) 10–20. [24] S. Ranganath, D. Liu, J.W. Sutherland A Comprehensive Model for the Flank Face Interference Mechanism in Peripheral Milling Trans. NAMRI/SME 26 (1998) 249–254. [25] J.A. Bailey, Friction in Metal Machining—Mechanical Aspects, Wear 31 (1975) 243-275. [26] J.H. Ko, W.S. Yun, D.W. Cho, K.F. Ehmann, Development of a Virtual Machining System, Part 1: Approximation of the Size Effect for Cutting Force Prediction, International Journal of Machine Tools and Manufacture 42(15) (2002) 1595-1605. [27] J.H. Ko, D.W. Cho, 3D Ball-End Milling Force Model Using Instantaneous Cutting Force Coefficients, ASME Journal of Manufacturing Science and Engineering 127 (2005) 1-12.
Highlights •End milling stability is predicted considering cross edge radiuses and flank edge profiles •The process damping effect is simulated according to the interactions between the edge profiles and undulation on the machined surface. • The fast and accurate model is proposed to simulate the process damping forces and the stability. • The simulation matches with the experimental phenomenon that tool wear causes high process damping.
25
PII: DOI: Reference:
S0890-6955(14)40016-6 http://dx.doi.org/10.1016/j.ijmachtools.2014.11.004 MTM3010
To appear in: International Journal of Machine Tools and Manufacture Received date: 30 August 2014 Revised date: 11 November 2014 Accepted date: 11 November 2014 Cite this article as: Jeong Hoon Ko, Time domain Prediction of milling stability according to cross edge radiuses and flank edge profiles, International Journal of Machine Tools and Manufacture, http://dx.doi.org/10.1016/j.ijmachtools.2014.11.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Time Domain Prediction of Milling Stability according to Cross Edge Radiuses and Flank Edge Profiles Jeong Hoon Ko* Singapore Institute of Manufacturing Technology, 71 Nanynag drive, 638075 Singapore * Corresponding author. Tel.: +65-6793-8592; fax: +65-6791-6377. E-mail. address:[email protected]
Abstract This article proposes a time domain model for predicting an end milling stability considering process damping caused by a variety of cross edge radiuses and flank profiles. The time domain model of calculating indentation areas, as well as regenerative dynamic uncut chips, is formulated for the prediction of the stabilizing effect induced by interference areas between the edge profiles and undulation left on a workpiece. The interference area generates forces against the vibration motion, which acts as a damping effect. In the model, the present and previous angular position of cross radiuses and flank edge profiles are located to calculate the dynamic uncut chip as well as indentation area based on a time history of the dynamic cutter center position. The phenomenon that chatter is damped according to cross edge radiuses and flank edge profiles is successfully simulated with the proposed dynamic model and validated through the extensive experimental tests. Keywords: time domain chatter model, cross-flank edge profiles, end milling stability, process damping
Nomenclature
hl
h f
local helix angle radial rake angle of cutting edge cutting edge angle helix angle flank elemental edge angle
Lf
flank elemental length
p
pitch angle
Fn (i, j , k ) normal pressure force Ff (i, j , k ) frictional force
n b Tc Nf
a Na
e
T ( ) c
unit vector tangent to the cutter edge unit vector normal to rake face unit vector on the rake surface and perpendicular to the cutter edge chip flow vector the number of flutes length of edge element along the radial direction. total disk number rotation angle of the cutter cutting edge location angle transformation matrix flute spacing angle
hc
uncut chip thickness
hcr
rescaled uncut chip thickness
ds Kn Kf θc Rr R er era rk
incremental cutting edge length normal cutting force coefficients frictional cutting force coefficients chip flow angle radial runout cutter radius cross edge radius separation angle cutting edge position from cutter center
1. Introduction Machining chatter often turns a great deterrent against high productivity and machining quality. There have been extensive research works on the mechanics and dynamics of milling processes. Tobias and Tlusty introduced first chatter stability laws in the frequency domain [1,2]. Sridhar presented the time domain solution with two coupled, delayed differential equations with time-varying coefficients [3]. Minis and Yanushevsky proposed the first analytical solution of the milling stability using Floquet's theory by advancing from Sridhar's formulation [4]. Altintas and Budak developed a general and closed solution of the milling stability in the frequency domain [5,6]. Some articles proposed added stability lobes to simulate the low radial immersion cutting during high speed machining [7,8]. Olgac et. al. included both single and multiple time delays and validated their stability law on uniform and variable pitch cutter results [9]. Ko and Altintas proposed mechanistic and dynamic models in time and frequency domain for plunge milling processes [10,11]. The developed stability model can predict a torsional-axial vibration as well as lateral vibrations. Eksioglu and Altintas [12] proposed a general formulation of flexible mills and workpiece at high axial and small radial depths of cuts. Recently, modeling of milling process damping has been improved by a few researchers [13-16]. Tunc and Budak [15, 16] proposed an indentation model based on the existence of a separation point on the cutting edge in order to estimate process stability. Eyniyan and Altintas [17, 18] proposed a dynamic cutting force model and a novel experimental technique for the identification of dynamic coefficients with a piezo actuator set-up. Researchers mentioned that the interaction between a wavy surface and a cutter flank should be further investigated as it affects the process damping. Regarding the edge radius effect, Waldorf et al. [19] proposed two ploughing models for static / orthogonal model and Endres [20] worked on the force balance under the deformation zone for static force prediction. Albrecht [21] investigated that the existence of hone radius causes the workpiece material to be extruded against the cutting tool, which increases process damping. Wu [22] investigated the relation of the penetration depth to the uncut chip thickness, vibration amplitude and the instantaneous shear angle. Shawky and Elbestawi [23] estimated the depth of penetration due to cross edge radius and Ranganath et al. [24] estimated the separation angle in order to incorporate indentation forces into the overall force model. Depending on tool wear or tool edge preparation, the profiles of the cross edge and its adjacent flank edge vary which affects the surface quality as well as stability significantly. Fig. 1 shows the examples of different cross edge radiuses according to insert or solid type of end mills. So, it is important to consider the cutting edge radiuses of the 2
tool as one of the variables for predicting machining stability. In addition, the profile of flank edge adjacent to cross edge may change according to tool wear, which need to be measured and modeled to simulate its damping effect accurately.
Fig. 1 cross edge measurement according to the type of cutting edge
This paper proposes a fast and comprehensive time domain model which simulates a dynamic uncut chip as well as the interaction of cross-flank edge profiles with the undulation of machined surface. It can predict how the edge profiles affect the milling stability by tracking the interaction between cross-flank edge profiles and undulation. Contact forces between cross–flank edges and wave on the machined surface contribute to the dynamics of the cutting process by increasing the overall damping. The elemental contact forces are integrated by simulating time-dependent elemental indentation volume between cross-flank edge and undulation profiles, which act against the cutting forces of removing the dynamic uncut chip. By tracking dynamic cutter center positions determined by vibration, runout, and feed motions, the time-dependent angular positions of the edge profiles as well as the undulation can be rapidly and precisely located. The angular positions are used to simulate the dynamic uncut chip as well as the contact area of cross-flank edge profiles against the undulation left on the machined surface. According to a variety of edge radiuses and flank profiles, the milling stabilities are experimentally compared and analyzed. The corresponding simulation results are presented and validated with the experimental ones on the damping effect with cross edge radiuses and flank edge profiles.
2. Time domain milling model considering cross-flank edge profiles Figure 2 illustrates the proposed time domain solution process. Firstly, after the cross edge radius and flank edge profiles are measured, tool geometry information, modal parameters, cutting conditions, and workpiece info are input to the simulation model. The edge profiles are digitized to simulate elemental forces and dynamic uncut chip thickness. After solving the dynamic equation using a numerical time integral scheme, the cutter center positions at 3
present and previous rotations are updated. By extracting the time history of the cutter center positions, the angular positions of the edge elements and the undulation on the surface can be estimated to formulate the dynamic uncut chip thickness as well as the intersection area between the edge profiles and the wave on the machined surface. The estimated dynamic uncut chip thickness and contact area are input to cutting and contact force model. Finally, the time-dependent cutting forces and vibrations can be continuously simulated using this procedure in order to evaluate the milling stability.
Fig. 2 Simulation procedure of time domain solution
The nose region of a cutting tool has a finite cross edge radius depending on tool edge preparation or wear. In front of the rounded portion of the tool nose, there is a separation angle (era ) which distinguishes the indentation from the shearing. The profile of flank edges may have curvature or different flank angles, which can be measured using optical devices and digitized into segmental flank angles. Fig. 3 shows the elemental cutting forces as well as indentation ones, which are integrated into total forces.
4
Fig. 3 unit vectors and force components component
The chip load and corresponding differential loads for each edge element are evaluated and digitally integrated to predict the total forces in three directions. The cutting forces acting on the rake surface of the disk element are divided into two orthogonal components: the normal pressure force, dFn (i, j , k ) , and the frictional force,
dF f (i, j , k ) . It can be obtained from
dFn (i, j , k ) K n ndAc (1)
dFf (i, j, k ) K f KnT c dAc where dAc h( ) cos r (a / cos hl ) , and n and T c are the unit vectors shown in Fig. 3, respectively [26].
cos cos hl T ( , hl ) sin cos hl sin hl
sin cos 0
cos sin hl sin sin hl cos hl
Using the transformation matrix considering local helix angle ( hl ) and each edge element’s angular location ( ),
dFn and dF f can be decomposed into three orthogonal force components in Cartesian coordinates as follows: dFx (i, j,k) [ K n (cos r cos cos hl sin r sin )
(2)
f 1 1 cos cos hl cos r sin 1 cos r cos sin hl ) f2 f2 2 r cos ) [ K n (cos r sin cos hl sinf K n K f cos c (sin r
dFy (i,j,k)
1 1 f1 1 f1 K cos cos cos r sin sin hl ) K f sin c (r f sin sin hl cos sin hl )] dA n KKf ncos c (sin r cos c f f 2f 2 f 2 2
K n K f sin c (
2
f1 1 cos sin sin hl )]dAc f2 f2
dFz (i, j,k) [ K n sin hl cos r K n K f cos c ( sin r K n K f sin c
1 cos hl ]dAc f2
f 1 sin hl 1 cos r cos hl ) f2 f2 5
The relationship between the rescaled uncut chip thickness hcr(i, j, k) and ln(Kn) can be readily derived using the Weibull function.
ln( Kn ) A1 ( A1 A2 )e( A3hcr )
A4
(3)
Kf and c can be represented as a nonlinear function dependent only on instantaneous uncut chip thickness hcr which is calculated using Eq. (4). K f B1 ( B1 B2 )e ( B3hcr )
c
B4
(4)
C1 C2 C2 1 (hcr / C3 )C4
The cutting force coefficients ( Kn, , Kf , and c ) can be identified using a mechanistic approach using a few cutting force measurements[26]. For example, the coefficients of Al 6061-T6 are calculated as A1 = 7.054, A2 = 10.303, A3 = 1.218, A4 = 1.040, B1 = 0.613, B2 = 0.912, B3 = 0.391, B4 = 2.772, C1 = 0.560, C 2 = 0.434, C3 = 4.548, and C 4 = 0.231. For S304, the coefficients of K n are defined as A1 = 7.467, A2 = 10.303, A3 = 1.218, A4 = 1.040. For estimating elemental contact forces which act as elemental process damping forces, the mean friction coefficient ( ) on the contact surface is assumed to be constant. And the normal force ( dPn ) and frictional force components ( dPf ) can be defined as follows:
dPn (i, j , k ) K d dV
(5)
dPf (i, j, k ) dPn where K d is the specific ploughing force over the unit volume dV of displaced work material, is the mean friction coefficient on the contact surface. K d and can be estimated using an iterative method based on the proposed time domain model compared to other approaches[15, 22, 25]. Considering the angular position ( ef ) of cross edge and flank profiles, the elemental indentation forces are transformed into Cartesian coordinates. The indentation is attributed to the actual contact between the workpiece and the tool nose region, which includes the tool cutting edge and its adjacent flank face. The indentation load and corresponding differential loads for each edge element are evaluated and digitally integrated to predict the damping forces acting against vibrations.
dFpx (i, j , k , f ) dPx ' cos(ef ) dPy ' sin(ef ) dFpy (i, j , k , f ) dPf sin(ef ) dPn cos(ef ) (6)
6
Here dPx ' and dPy ' for flank edges are defined as follows:
dPx ' dPff cos f dPnf sin f
(7)
dPy ' dPff sin f dPnf cos f For cross edges, the force components are defined as follows:
dPx ' dPfc cos c dPnc sin c
(8)
dPy ' dPfc sin c dPnc cos c
In the elemental indentation force components, c changes according to the type of tooling as well as tool wear. Since the flank edge may have curvature, the flank angle f may have different values according to elemental flank edge index f .
Finally, total forces can be defined as follows
F(j) dF (i, j, k ) dFp (i, j, k , f ) i
k
i
k
f
(9)
The dynamic equation for milling system can be formulated as follows:
x xx y yx z zx
xy yy zy
xz Fx yz Fy zz Fz
(10)
where M is modal mass, C is modal damping, K is modal stiffness. Coordinate ab ( a or b is x, y, z or ) denotes the displacement of the cutter center coordinate a when cutting force F is applied in direction b. From experimental modal analysis, the direct transfer function of the structure along the x, y, z direction at the tip of the end mill is obtained in the Laplace domain as,
ab (s)
wnh 2 / kh a Nh 2 Fb h1 s 2 h wnh s w2nh
(11)
where Nh is the total number of modes in the system, h represents each of these modes and wnh , k h , and h are the natural frequency, modal stiffness, and damping ratio, respectively. The fourth order Runge-Kutta equation is used for the numerical integration of the differential Eq. (10). The total displacement of the tool is evaluated by summing the vibrations contributed by all natural modes. Nh
a (t ) a h ( t )
(12)
h 1
7
Finally, the center position of the end mill ( X c , Yc , Z c ) can be described using the vibration displacement and feed per tooth ( f t ) as follows:
X c (t ) feed x t Rr sin x(t ) Yc (t ) feed y t Rr cos y (t ) Z (t ) z (t ) c feed z t Cutter Center
Rigid Body Motion
(13)
Vibration
where Rr represents the radial runout of the cutter center.
3. Estimation of dynamic uncut chip and indentation area This section explains the fast and accurate simulation approach on dynamic uncut chip thickness as well as flankwave interaction volume. The proposed simulation method takes less than 1.5 Sec for predicting the process of one cutting condition with a simulation time interval digitized at least ten times smaller than the period of the highest frequency among modal frequencies.
3.1 Previous and present cutting edge positions on vibration wave Firstly, the cutting edges including cross and flank edges are divided into a finite number of small differential elements in the cutting edge curve. The angular position e (i, j , k ) of the kth edge element of the tooth i at the jth angular position of the cutter is described by
e (i, j, k ) r ( j ) (i 1) p
k da tan h , r ( j ) j r R
(14)
where r is the rotation angle of the cutter measured from y axis, and p 2 / N is the pitch angle. In the proposed model, the present and previous edge positions can be calculated using the present and previous cutter center positions and cutter edge geometry information. Using Eq. (10), the cutter center position ( X c , Yc , Z c ) is updated and saved according to incremental angular value r corresponding to the simulation time interval considering the period of highest natural frequency. Since the cutter center positions are saved according to cutter rotation angle , the previous cutter center positions ( X cm , Ycm , Z cm ) can be extracted by locating the previous cutter rotation angle m . Fig. 4 summarizes the overall algorithm of dynamic uncut chip thickness calculation method by using previous and present cutter center positions.
8
(a) Angular mismatch
v
(b)
e
has the same value as
em
(c) Algorithm of tracking present and previous cutting edge for dynamic uncut chip thickness
Fig. 4 Dynamic uncut chip thickness calculation algorithm
9
In order to consider the phase of the vibration wave, the dynamic uncut chip thickness is defined as the distance between the present and previous cutting edge locations which are aligned to the radial direction of the present carter center. To calculate the radial directional dynamic uncut chip thickness, the previous cutting edge’s angular location (
em ) is to match with the present cutting edge’s angular location ( e ). In the case that e has the different value as em over the tolerance (ε) as shown in Fig. 4(a), v ( e - em ) should be added to previous cutter center location (
cm ) in order to relocate previous cutting edge. Finally, when e has the same value as em within the tolerance as in Fig. 4(b) after a few iterations, two cutting edges’ distance is calculated as dynamic uncut chip thickness. The algorithm incorporates the time delay of the vibration phase by accurately locating two cutting edges forming vibration wave as explained in Fig. 4(c). The coordinates of the points along the cutting edge are evaluated by using the cutter geometry information. The coordinates ( Xe, Ye , Ze ) of a point along the cutting edge of tooth j are expressed as:
X e, j (t ) X c (t ) rk sin( j ), Ye, j Yc (t ) rk cos ( j ), Ze,j (t ) Z c Raj (rk l ) tan r Where
(15)
rk ( X e X c )2 (Ye Yc )2 j t ( j 1) p
So the possible dynamic uncut chip thickness is evaluated by subtracting the radial distance of the present teeth from the previous one with reference to the cutter center as follows:
hkm ( j (t )) ( X e X c )2 (Ye Yc )2 ( X em X c )2 (Yem Yc )2
(16)
The final dynamic uncut chip thickness is selected as the minimum value among hkm and should be larger than zero as follows:
h(i, j , k ) Max[0, Min( hkm (i, j, k , m))]
(17)
where hkm(i, j, k, m) represents the possible uncut chip thicknesses. m is previously passed teeth (m = 1 , …, Nf) at the same edge location angle as the present angle and h indicates the real uncut chip thickness.
3.2 Interaction of cross-flank edges against undulation of the machined surface In order to consider the indentation volume between cross-flank edge profile and the undulation left on the surface, the cross and flank edges are divided into a finite number of small differential elements. In Fig. 5(a), the angular position, ef of the lth cross-flank element of the kth disk of the tooth i at the jth angular index is defined by
ef (i, j, k , f ) r ( j ) (i 1) p
k da tan h f (l ) R
(18)
where f (l ) , the angular position of the cross-flank edge element is estimated as follows.
10
f (l ) cos 1 (
dey (l ) dex 2 (l ) dey 2 (l )
)
(19)
Where dex (l ) l df , dey (l ) R er (er 2 (l df )2 ) for the section of cross edge radius er and l = 0.. n1. The
position
of
the
elemental
flank
edges
is
defined
as
dex (l ) (l n1 ) df cos( f )
,
dey (l ) (l n1 ) df sin( f ) where l = n1+1…n2. Here, n2 is the total digitization number of cross and flank edges. As in Fig. 5(b), flank edges may have curvature or may have different angles, which are divided into smaller segment with various angles ( f 1 , f 2 , f 3 , .. and so on ) according to the profile. The angles can be measured and identified from 3D optical devices. So, depending on the profile of the cross and flank edges, dex and de y can be defined accordingly. The initial previous cutting edge location is estimated as follows.
elm (i, j, k ) mN r (i 1) p , r ( j ) j r , N 360 /
(20)
Then a few iterations are required until its difference ( v ) between ef and elm reduces less than the tolerance ( ) according to the algorithm explained in Fig. 5(c). Finally, the intersection height is calculated as follows
dI h max[0, min( Pef Pelm )] where Pelm ( X elm X c )2 (Yelm Yc )2 and
Pef ( X ef X c )2 (Yef Yc )2
(21) Depending on the undulation of the workpiece and the profile of cross and flank edges, the indentation height has zero or multiple potential values. Its minimum value over zero is the actual indentation height. After this indentation height is estimated, the elemental indentation volume ( dV dI h da df ) are input to the Eq. (5) to calculate the process damping forces. The formation enables the prediction of the stabilizing effect induced by the indentation areas between cross-flank edges and undulation left on machined surfaces according to cross edge radiuses and flank edge profiles.
11
(5-b) cross edge and its adjacent flank (5-a) Calculation of elemental indentation areas on cross and flank edges
angles
(5-c) Algorithm for calculating indentation areas between flank edge components and undulation on the surfaces
Fig. 5 Estimation of angular position of the cross edge and adjacent flank edge elements
4. Comparison of experimental and predicted milling stabilities 4.1 Overall stability changes according to cross-flank edge profiles The profiles of cross-flank edge profiles were generated by gradually wearing out sharp cross edges through 12
repetitive machining. 3D non-contact metrology system was used to measure the accurate values of the edge radiuses as well as flank elemental angles. The values were averaged from a series of measurements of cross edges and digitized into the simulation model. In this paper, three different CERs and flank profiles were generated and categorized as fresh tool, worn tool-1 and worn tool-2 as shown in Fig. 6. With the three sets of the edge profiles as described in Table 1, the milling stability tests were performed and compared over various RPMs and cutting depths for the work materials such as Al6061-T6 and SUS304 as listed in Table 1. Around more than 80 tests were performed so that the predicted milling stabilities can be compared with measured ones according to the different edge profiles.
13
(6-a) Fresh Tool
(6-b) Worn Tool-1
(6-c) Worn Tool-2 Fig. 6. Changes of cross edge radiuses according to tool wear
14
Machine
Roeders milling machine Dia. = 10 mm, No. of flute = 2, Rake angle = 10°, Helix angle = 30°, flank angle = 10° CER = 8.168 µm, f 1 =7.391° L f 1 =100.37 µm Fresh Tool CER = 23.349 µm, f 1 =-24.940° L f 1 =32.247
Milling Cutters with different CER (Cross Edge Radius)
µm, f 2 = 6.5863°, L f 2 =63.763 µm, f 3 = Worn Tool-1
7.2821°, L f 3 =50.537 µm CER = 32.690 µm, f 1 =-11.467° L f 1 =31.115 µm, f 2 = 1.583°, L f 2 =21.214 µm, f 3 =
Worn Tool-2 Cutting Conditions
4.076°, L f 3 =56.617 µm
Feed / tooth = 0.05, AE = 3 mm, AP = 2 mm to 4 mm RPM range from 5000 to 11500 RPMs
Materials Tested AL6061-T6 and SUS 304 Table 1. Experimental milling conditions ( The runout of the tools: 0.004 mm ) For the side milling processes, the modal parameters are experimentally taken for x and y directions as shown in Table 2. The listed natural frequency, damping and stiffness composes the transfer function ab of the tooling on the spindle as defined in Eq. (11). The identified transfer function ab is input to the model as indicated in Eq. (10) in order to simulate the dynamic movement of the end mill. As seen in modal parameters, it is expected that a chatter frequency would be around 4.8 KHz since the dynamic stiffness of the mode no. 1 is least compared to other modes listed in Table 2. Table 2. Measured modal parameters of the end mill attached to Roeders 760 Mode no. xx
yy
Natural Frequency (
Damping Ratio ( )
Hz)
Modal Stiffness (k ) (N/m)
1
4800
0.014
15659818
2
1062.5
0.012
64108769
3
4362.5
0.013
77923634
4
2068.75
0.038
35180967
1
4812.5
0.014
18061694
2
1068.75
0.012
54989075
3
4368.75
0.010
94097720
4
1931.25
0.044
37353541
Figs. 7 and 8 illustrate the changes of overall stability lobes over tested conditions with the comparisons of measured and predicted results. The effects of cutting conditions on the milling stability are clearly compared 15
according to edge profiles of each tool. The experimental machining stability is validated through machined surface quality and sound signal’s FFT. For example, the tested conditions are categorized as “stable” conditions if it dampens the chatter mode with 4.8 KHz in FFT of sound signals and produces surface roughness less than Ra 0.5 µm without chatter mark. As shown in Figs. 7 and 8, the milling stabilities vary according to different tools, which means that cross-flank edge profiles play a critical role in affecting process stability. Since worn tool-2 has a higher CER than other tools, it is understood that as CER increases, the milling process becomes stable due to increased process damping.
(a) Stabilities with fresh tool
(b) Stabilities with worn tool-1
(c) Stabilities with worn tool-2 Figure 7 Changes of stabilities in experimental and simulation results according to cross-edge profiles for milling Al6061-T6 16
(a) Stabilities with fresh tool
(b) Stabilities with worn tool-1
(c) Stabilities with worn tool-2 Figure 8. Changes of stabilities in experimental and simulation results according to cross-edge profiles for milling SUS 304
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4.2 Comparison and analysis of measured and predicted stabilities according to cross-flank edge profiles In Fig. 9(a), clear chatter marks are shown in the photo of the machined surface with fresh tool while Fig. 9(b) illustrates clean surface with worn tool 2. Accordingly, the sound signal’s FFT illustrates that the chatter frequency around 4.8 KHz appears with the fresh tool as shown in Fig. 9(c) while it is suppressed with the damping effect of the worn tool-2 as in Fig. 9(d).
(9-a) Ra = 2.501µm for new tool
(9-b) Ra = 0.281 µm for worn tool-2
(9-c) Sound signal’s FFT for new tool
(9-d) Sound signal’s FFT for worn tool-2 Figure 9. Comparison of measured surface roughness and sound signals for milling SUS 304 with fresh tool and worn tool-2: RPM 8000, AE 3 mm, AP 2 mm, Feed rate 800mm/min
By inputting the geometry and cutting condition into the proposed time domain model, the dynamic uncut chip is calculated and cutting forces are predicted considering the cutting force coefficients of test materials. For the flankwave contact force coefficients, K d = 22000 N/mm3 and = 0.3 are used for AL6061-T6. K d and are defined as 56000 N/mm3 and 0.3 respectively for SUS304. A separation angle (era ) is identified as 17° for the simulation. As shown in Fig. 10(a), the simulated forces are showing chatter phenomenon for end milling with the fresh tool for the given condition at RPM 8000. However, the oscillating forces with chatter frequency 4.8 KHz are damped out in end milling with worn tool 2 as in Fig. 10 (b). (Please note that the dynamometer used in the experiment has an integrated filter 200 Hz, which is not able to provide measured forces comparable to predicted ones.) In addition, the cutter 18
center position’s movement is simulated at RPM 8000, and FFTs of the simulated displacements are analyzed. The chatter vibration with worn tool-2 is damped as illustrated in Fig. 10(d) while it appears at FFT of displacements for end milling with the fresh tool in Fig. 10(c). So the simulation results have good agreements with the experimental ones in Fig. 9 in terms of the milling stability according to edge profiles.
(10-a) Simulated forces for fresh tool
(10-c) FFT of simulated displacements for fresh tool
(10-b) Simulated forces for worn tool 2
(10-d) FFT of simulated displacements for worn tool-2
Fig. 10 Simulated forces and FFTs of cutter displacements for milling SUS 304 with fresh tool and worn tool-2 for condition RPM 8000, AE 3 mm, AP 2 mm, Feed rate 800mm/min
For the end milling of Al 6061-T6 at RPM 11500, the measured surfaces and FFT of sound signals are compared according to the fresh tool and the worn tool-2 as shown in Fig. 11. It is clear that worn tool-2 causes higher process damping than the fresh tool as the chatter frequency is damped, and the surface roughness is better with worn tool-2.
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(11-a) Ra = 0.470 µm for fresh tool
(11-c) Sound signal’s FFT for fresh tool
(11-b) Ra = 0.196 µm for worn tool-2
(11-d) Sound signal’s FFT for worn tool-2
Figure 11. Comparison of measured surface roughness and sound signals for milling Al6061-T6 with fresh tool and worn tool-2: RPM 11500, AE 3 mm, AP 2 mm, Feed rate 1150 mm/min According to the time domain simulation for the same cutting condition as in Fig. 11, the simulated cutting force profiles are compared with different tools as illustrated in Figs 12(a) and (b). And the FFT of the displacements of the simulated cutter centers shows different stabilities according to fresh and worn tool-2 in the Figs 12(c) and (d), which match with the experimental stabilities as illustrated in Fig. 11.
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(12-a) Simulated forces for fresh tool
(12-c) FFT of simulated displacements for fresh tool
(12-b) Simulated forces for worn tool-2
(12-d) FFT of simulated displacements for worn tool-2
Figure 12. Comparison of predicted forces and FFTs of cutter displacements for milling Al6061-T6 with fresh tool and worn tool-2: RPM 11500, AE 3 mm, AP 2 mm, Feed rate 1150 mm/min
Fig. 13 illustrates the change of FFT of measured sound signals according to three tools at RPMs 8000 and 11500. It is observed that RPM 8000 incurs more process damping than RPM 11500 considering that, in end milling with worn tool-1, the chatter frequency around 4.8 KHz is damped at RPM 8000 while the chatter frequency still appears at RPM 11500. In the case of the fresh tool, the chatter frequencies appear at both RPMs 8000 and 11500 while the frequencies are damped out at these RPMs with worn tool-2. The same phenomenon appears in predicted results as well in Fig. 14 which shows FFT of cutter center’s displacements according to RPMs 8000 and 11500. Especially, at RPM 8000, the chatter frequency around 4.8 kHz is damped with worn tool-1 but the frequency still appears at RPM 11500. So it is well validated that the time domain model can predict the effect of cross-flank edge profiles on process damping at different RPMs.
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fresh tool (chatter)
worn tool-1 (stable)
worn tool-2 (stable)
(a) Stability change according to different edge profiles at 8000 RPM based on FFTs of sound signals
fresh tool (chatter)
worn tool-1 (chatter)
worn tool-2 (stable)
(b) Stability change according to different edge profiles at 11500 RPM based on FFTs of sound signals Figure 13. Comparison of FFT of the measrued sound signals for milling SUS 304 : RPMs 8000 and 11500, AE 3 mm, AP 2 mm, feed per tooth 0.05 mm/tooth
fresh tool (chatter)
worn tool-1(stable )
worn tool-2 (stable)
(a) Stability change according to different edge profiles at 8000 RPM based on FFT of predicted displacements
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fresh tool (chatter)
worn tool-1(chatter)
worn tool-2(stable)
(b) Stability change according to different edge profiles at 11500 RPM based on FFT of predicted displacements Figure 14. Comparison of FFT of simulated cutter displacements for milling SUS 304 for condition RPMs 8000 and 11500, AE 3 mm, AP 2 mm, feed per tooth 0.05 mm/tooth
5. Conclusion This article proposes the time domain model for predicting the end milling stability according to cross-flank edge profiles over various cutting conditions. The model simulates the interaction areas between cross-flank edge profiles and undulation ones of the machined workpiece, which produces a damping against regenerative vibrations. The time-dependent angular positions of the edge profiles can be accurately located according to the undulation profiles left on workpieces using present and previous dynamic cutter center position determined by vibration and rigid body motion. By dynamically tracking the cutter movements, it predicts overall dynamic forces and vibration generated by the edge profiles against the wavy machined surface. The phenomena that chatter is damped according to the various cross edge radiuses and adjacent flank profiles were successfully simulated with the proposed model and experimentally validated through the extensive experimental tests. The simulation and experimental results of the milling stabilities with the fresh tool, worn tool-1 and worn tool-2 were compared over various cutting conditions for the end milling of Al6061-T6 and SUS304. The chatter frequency 4.8 kHz still appeared at end milling with fresh tool at the tested RPMs, but it was damped with worn tool-1 at spindle RPMs less than 8000 RPM. If RPM increased to 11500 RPM, the frequency appeared even with worn tool-1. However, the end milling with worn tool-2 has a strong process damping effect, which damps the chatter frequency even at 11500 RPM. So, for the variety of the cross edge radiuses and adjacent flank profiles, the machining stability can be accurately predicted using the proposed model. It is expected that the simulation approach will be used to design cross-flank edge geometry for the stabilization of the milling chatter, which will be further investigated in the near future. The research will be useful for a tooling industry regarding the design of the tool geometry. In terms of the tool design, the cross-flank edge profiles have a critical role in terms of the stability as well as the surface quality. Depending on the surface requirement as well as operating parameters, the cross-flank edge profile needs to be optimized so that the machining stability may be secured for the required machined surface quality.
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Acknowledgements The author appreciates support from Mr. Ng Thai Ee and Mr. Ruben S/O Sukumar from SIMTech. References [1] S.A. Tobias SA, W. Fiswick, Theory of Regenerative Machine Tool Chatter, Engineering, London (1958) 258. [2] J. Tlusty, M. Polacek, Examples of the Handling of the Self-excited Vibration of Machine Tools, FoKoMa, Hanser publisher, Munich (1957) [3] R. Sridhar, R.E. Hohnand, G.W. Long, General Formulation of the Milling Process Equation, ASME Journal of Engineering for Industry 90 (1968) 317-324. [4] I. Minis, T. Yanushevsky, A New Theoretical Approach for the Prediction of Machine Tool Chatter in Milling, ASME Journal of Engineering for Industry 115 (1993) 1-8. [5] E. Budak, Y. Altintas, Analytical Prediction of Chatter Stability in Milling-Part I: General Formulation, ASME Journal of Dynamic Systems, Measurement, and Control 120 (1) (1998) 22-30. [6] E. Budak, Y. Altintas, Analytical Prediction of Chatter Stability in Milling- Part II: Application of the General Formulation to Common Milling Systems, ASME Journal of Dynamic Systems, Measurement, and Control 120 (1) (1998) 31-36. [7] T. Insberger T, G. Stepan, Stability of the Milling Process, Periodica Polytechnica- Mechanical Engineering 44 (2000) 47-57. [8] M.A. Davies,
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[16] L.T. Tunc, E. Budak, Identification and Modelling of Process Damping in Milling, ASME Journal of Manufacturing Science and Engineering135(2) ( 2013) [17] Y. Altintas, M. Eynian, H. Onozuka , Identification of dynamic cutting force coefficients and chatter stability with process damping , Cirp Annals-manufacturing Technology 57(1) 2008 371-374. [18] M. Eyniyan, Y. Altintas, Analytical Chatter Stability of Milling with Rotating Cutter Dynamics at Process Damping Speeds, ASME Journal of Manufacturing Science and Engineering 132(2) (2010) 1-14. [19] D.J. Waldorf, R.E. Devor, S.G. Kapoor, An Evaluation of Ploughing Models for Orthogonal Machining,” ASME Journal of Manufacturing Science and Engineering 121 (1999) 550–558. [20] J. Manjunathaiah, W.J. Endres A New Model and Analysis of Orthogonal Machining With an Edge-Radiused Tool ASME Journal of Manufacturing Science and Engineering 122 (2000) 384–390. [21] P. Albrecht New Development in the Theory of Metal-Cutting Process, Part I. The Ploughing Forces in Metal Cutting ASME Journal of Engineering for Industry 82 (1960) 348–358. [22] D.W. Wu Application of a Comprehensive Dynamic Cutting Force Model to Orthogonal Wave-Generating Processes International Journal of Mechanical Sciences 30(8) (1988) 581–600. [23] A.M. Shawky, M.A. Elbestawi, An Enhanced Dynamic Model in Turning Including the Effect of Ploughing Forces ASME Journal of Manufacturing Science and Engineering 119(1) (1997) 10–20. [24] S. Ranganath, D. Liu, J.W. Sutherland A Comprehensive Model for the Flank Face Interference Mechanism in Peripheral Milling Trans. NAMRI/SME 26 (1998) 249–254. [25] J.A. Bailey, Friction in Metal Machining—Mechanical Aspects, Wear 31 (1975) 243-275. [26] J.H. Ko, W.S. Yun, D.W. Cho, K.F. Ehmann, Development of a Virtual Machining System, Part 1: Approximation of the Size Effect for Cutting Force Prediction, International Journal of Machine Tools and Manufacture 42(15) (2002) 1595-1605. [27] J.H. Ko, D.W. Cho, 3D Ball-End Milling Force Model Using Instantaneous Cutting Force Coefficients, ASME Journal of Manufacturing Science and Engineering 127 (2005) 1-12.
Highlights •End milling stability is predicted considering cross edge radiuses and flank edge profiles •The process damping effect is simulated according to the interactions between the edge profiles and undulation on the machined surface. • The fast and accurate model is proposed to simulate the process damping forces and the stability. • The simulation matches with the experimental phenomenon that tool wear causes high process damping.
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