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An accuracy algorithm for chip thickness modeling in 5-axis ball-end finish milling
Computer-Aided Design 43 (2011) 971–978
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An accuracy algorithm for chip thickness modeling in 5-axis ball-end finish milling Xin-Guang Liang a,∗ , Zhen-Qiang Yao a,b a
School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China
b
State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, PR China
article
info
Article history: Received 26 October 2010 Accepted 25 April 2011 Keywords: Chip thickness 5-axis mill Ball-end finishing 3D trochoidal tooth trajectory Engagement boundary
abstract Chip thickness in milling is one of the most fundamental parameters, which can significantly affect cutting force, cutting heat, cutting stability and machined surface topography for computer-aided process planning. In this paper, a combination of a three-dimensional trochoidal tooth trajectory model (3D3T) and engagement-boundary chip model is developed to determine instantaneous chip thickness in 5-axis ball-end finish milling. In comparison with the chip volume measured in a commercial software package (Unigraphics) the accuracy of the proposed model has been numerically validated with various process parameters including cutting depth, tool–workpiece inclination and cutter runout. The differences in time-varying delay and dynamic chip thickness as well as stability are compared with different models to show the impact of using 3D3T mechanism for chip thickness modeling in 5-axis ball-end finish. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction 5-axis ball-end milling of 3D free-form surfaces such as dies, molds, turbines and various aerospace components is gaining wider applications. Chip thickness, one of the most fundamental parameters in milling process planning, can significantly affect cutting force, cutting heat, cutting stability and machined surface topography. Over the past 20 years, chip thickness modeling in 5-axis ball-end milling has been receiving great attention. In general, most chip thickness models were more or less dependent on two basic assumptions. One is the hemisphere paths assumption (HPA) that trajectories of cutting tooth are decomposed into pure translation of tool axis and revolution around the tool axis in series, and spheric rules can be transplanted expediently. The other can be regarded as the sine product assumption (SPA) in which the chip thickness model is simplified as Ct = f sin ϕ sin k, namely, feed per tooth f , revolution angle ϕ and immersion angle k become three decoupled parameters. Both assumptions can make it more convenient and efficient to analytically model chip geometry, cutting force and stability in ball-end milling [1–7]. Unfortunately, with the development of research both HPA- and SPA-induced model errors have been detected either in axial or in circumferential, especially for lightcut finishing where cutting depth, width and feed are all within 10% of cutter radius.
∗
Corresponding author. Tel.: +86 21 3420 6583. E-mail addresses: [email protected], [email protected] (X.-G. Liang). 0010-4485/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cad.2011.04.012
Many efforts have been involved in using the methods of 3D grid descriptions or constructive solid geometry (CSG) and indeed improved modeling accuracy of chip thickness. Altintas and Lee [8] proposed a Z -map method to eliminate the circumferential error of sin ϕ in SPA. Zhu et al. [9] proposed a distance method which needs to calculate the intersection of the previous tooth envelope surface and a vector line passing through the instantaneous ball center. Although this model can remove the whole error induced by SPA, the intersection point would be much more difficult to obtain directly without HPA. Sun et al. [10] proposed a 2D trochoidal tooth paths method to calculate the relative chip thickness in 3-axis ballend milling, which can partly get rid of HPA. However, it was not mentioned how to deal with the inevitable axial error of SPA when determining the definite chip thickness. Roth et al. [11] proposed an adaptive depth buffer method to save computing memory by sizing the depth buffer to the tool as opposed to the workpiece. Khachan and Ismail [12] furthered this solid graphics method by describing the chip thickness in the form of chip volume, and simulated machining chatter in 5-axis milling. This method can be extended to the true cutting path as trochoid and thoroughly eliminate chip thickness errors induced by SPA and HPA. However, it did not clearly describe how to determine the non-uniform delays between adjacent teeth. Fixing the uniform delays to the period of tooth can cause different cutting force and dynamics. Li et al. [13] applied Taylor’s series to express the non-uniform delays and revealed the errors of traditional model in 3-axis milling. Faassen et al. [14] discussed the influences of non-uniform delays on the shift of stability lobes in 3-axis milling of low radial immersion.
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Besides the two assumptions, engagement region filter (ERF) is another simplified method commonly used to determine the activity of chip thickness. As long as tooth segments locate in engagement region, the corresponding chip thicknesses would be activated by multiplying one, or else they will be replaced by zeros [6,15–18]. However, ERF can result in discontinuous steps of chip thickness around engagement boundaries, making chip thickness always overestimated. The overestimated chip thickness can also become more significant in finish milling. Until now, there have been few existing models that can completely step out of the two assumptions and ERF simplification, especially in 5-axis milling. This paper presents a three-dimensional trochoidal tooth trajectory (3D3T) model to describe the definite chip thickness for 5-axis ball-end milling in finishing processes without using the assumptions HPA and SPA. An engagement boundary chip model is also established to eliminate the overestimated chip thickness error efficiently. Based on the fact that chip volume can directly reflect the accuracy of chip thickness [12], numerical chip volume generated from the proposed model and a commercial software package are compared to validate the accuracy of the proposed model by studying the effects of cutting depth, tool inclination and cutter runout. Impacts of the proposed model on milling stability and stable vibration cycle are mentioned for future applications.
Fig. 1. Cutter geometries and runout definitions.
2. Chip thickness modeling with 3D3T 2.1. Cutter runout and 3D3T in 5-axis milling A schematic representation of cutter geometry and its runout is shown in Fig. 1. Note that R denotes the radius of the cutter and l0 represents helix angle. Point P located on the ith flute, has elevation ZT , revolution radius r and lag angle Ψ . The lag angle Ψ is given by Eq. (1).
ψ(ZT ) = (R + ZT ) tan l0 /R.
Due to the presence of cutter runout, the tool coordinate system OXT YT ZT and the spindle coordinate system OS XS YS ZS no longer coincide with each other; their relative positions can be described by runout parameters e and ρ as shown in Fig. 1. Cutting radius, Rti ; spherical radius, RSi ; immersion angles, ki (ZT ) as well as cutter runout induced lag angle βi (ZT ) can be updated in the spindle coordinate system OS XS YS ZS using Eqs. (2)–(5), respectively where Yi (ZT ) = ρ + Ψ (ZT ) + 2π (i − 1)/n. Rti = RSi =
ri (ZT )2 + e2 − 2ri (ZT )e cos γi (ZT )
Rti2 + ZT2
ki (ZT ) = cos
−1
(ZT /RSi )
βi (ZT ) = [sign(sin γi )
sign(sin γi )
Fig. 2. Cutting orientations in 5-axis milling.
(1)
] cos [(
Rti2
1 0 0
0 cos(t ) sin(t )
0 − sin(t ) cos(t )
cos(l′ ) 0 − sin(l′ )
0 1 0
sin(l′ ) 0 . cos(l′ )
(2)
TM =
(3)
Therefore, parameters of tooth trajectory in OXN YN ZN and process parameters in OS XS YS ZS can be transformed from Eq. (10). Two 3D trochoidal tooth trajectories, corresponding to two distinguished points on the same cutting edge are depicted in Fig. 3.
(4) −1
In the 5-axis ball-end milling case, the tool axis is no more orthogonal to the feed direction due to the tool–workpiece inclination. For an arbitrary linear feed portion of the whole tool trajectory, the involved coordinate systems are depicted in Fig. 2. A transformation from OS XS YS ZS to OXN YN ZN , denoted by TM , can be derived from Eq. (9) where l and t denote lead angle and tilt angle respectively, and l′ = tan−1 (tan(l) cos(t )).
+ ri − e )/(2Rti ri )]. (5) 2
(9)
Combining the influence of cutter runout and flute helix, the total lag angle δi (ZT ), which can be used to determine the actual revolution angle of the cutting edge, is calculated using Eq. (6).
δi (ZT ) = βi (ZT ) + ψ(ZT ).
In 3D3T modeling, instantaneous uncut chip thickness can be calculated through the instantaneous position of the cutter center as depicted in Fig. 3. Point M is the instantaneous location on the ith cutting edge of ZT , point N denotes the point on the dth previous cutting edge; point Oi and point Oi_d represent their corresponding ball centers respectively. Additionally, line Oi M just passes by point N. τi_d (t ) is defined as the delay that is involved when calculating the chip thickness that tooth i removes at time t with respect to the tooth i − d. For example as shown in Fig. 3, τ2_1 (t ) is the difference between the time when tooth 2 is at point M and
(6)
Then, the revolution angle ϕi (ZT , t ) can be given by Eq. (7).
ϕi (ZT , t ) = ϕi (−R, 0) + ωt − δi (ZT ).
(7)
The flutes interval angle for a cutter of n flutes can be expressed by Eq. (8), where ω and τˆ are defined as spindle angular speed and pitch of tooth respectively.
ωτˆ = ϕi−1 (−R, 0) − ϕi (−R, 0) = 2π /n.
(8)
XN YN ZN
XT YT ZT
= TM
XT YT ZT
or
XN YN ZN
′
= TM
.
(10)
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B = Ct2_1 cos(ϕ2 (ZT _r2 , t )) sin(k2 (ZT _r2 )) = RS2 cos(ϕ2 (ZT _r2 , t )) sin(k2 (ZT _r2 )) − RS1 cos(ϕ1 (ZT _r1 , t − τ2_1 (t ))) sin(k1 (ZT _r1 )) + fTM (1, 2)τ2_1 (t )/τˆ
(14)
C = Ct2_1 cos(k2 (ZT _r2 )) = RS2 cos(k2 (ZT _r2 )) − RS1 cos(k1 (ZT _r1 )) − fTM (1, 3)τ2_1 (t )/τˆ .
(15)
The local geometry parameters of point M at time t in the above three equations, including k2 , ϕ2 , ZT _r2 and RS2 , are easy to express as input parameters. Each equation can directly formulate the chip thickness. However, it will result in an error when the denominator of any of these equations approaches zero. Therefore, the Pythagorean proposition is applied twice to eliminate the parameter ϕ2 and k2 on the left, yielding collection D in Eq. (16) and instantaneous chip thickness Ct in Eq. (17) which can always be validated whatever ϕ2 or k2 is getting zero or not.
Fig. 3. (a) 3D3Ts and (b) previous cutting surface descriptions.
the time when tooth 1 is at point N. Because feed per tooth is always rather smaller compared with the dimension of the sculpture feature on the workpiece surface, linear feed assumption can be adopted within the scale of feed per tooth. The position of point N on tooth 1 can be described as a sum of the position of the instantaneous ball center O1 and the position of cutting edge relative to O1 at time t − τ2_1 (t ). The expression is given by Eq. (11). XT _N1 YT _N1 ZT _N1
XT _c1 YT _c1 ZT _c1
=
XT _r1 + YT _r1 ZT _r1
ϕ0 (t − τ2_1 (t ))f /(ωτˆ )
′
0 0
= TM +
RS1 sin(k1 (ZT _r1 )) sin(ϕ1 (ZT _r1 , t − τ2_1 (t ))) RS1 sin(k1 (ZT _r1 )) cos(ϕ1 (ZT _r1 , t − τ2_1 (t ))) . RS1 cos(k1 (ZT _r1 ))
(11)
XT _N2 YT _N2 ZT _N2 ′
XT _c2 YT _c2 ZT _c2
=
XT _r2 + YT _r2 ZT _r2
(17)
where θi_d (ZT _ri ) = δi−d (ZT _r (i−d) ) − δi (ZT _ri ). The corresponding delay τ2_1 and previous tooth elevation ZT _r1 can be solved firstly by either numerically iterating Eqs. (18) and (19) for an exact solution or linearizing them for an approximate solution. When initial conditions τ2_1 = 0 and ZT _r1 = ZT _r2 are used for iteration with the Newton–Raphson method, exact (99.9% approximate) solutions can be obtained within two or three iterations for most of the points on the previous machined surface as shown in Fig. 4. H1 (ZT _r1 , τ (t )) = A cos(ϕ2 (ZT _r2 , t )) − B sin(ϕ2 (ZT _r2 , t )) = 0 (18) (19)
Then, the 3D3T-based instantaneous chip thickness for an arbitrary tooth i can be numerically derived from Eq. (20). The algorithm can also be explained in Fig. 5. d = 1, 2, . . . , n (20) where Ct i_d denotes the static chip thickness between arbitrary tooth i and tooth i − d, as expressed in Eq. (21). i_d = RSi − RSi−d [cos(ki−d (ZT _r (i−d) )) cos(ki (ZT _ri )) Ct
0 0
(RS2 − Ct2_1 ) sin(k2 (ZT _r2 )) sin(ϕ2 (ZT _r2 , t )) + (RS2 − Ct2_1 ) sin(k2 (ZT _r2 )) cos(ϕ2 (ZT _r2 , t )) . −(RS2 − Ct2_1 ) cos(k2 (ZT _r2 ))
Ct2_1 = C cos(k2 (ZT _r2 )) + D sin(k2 (ZT _r2 )) = RS2 − RS1 [cos(k1 (ZT _r1 )) cos(k2 (ZT _r2 )) + sin(k1 (ZT _r1 )) sin k2 (ZT _r2 ) cos(ωτ2_1 (t ) − ωτˆ + θ2_1 (ZT _r2 ))] + f {[TM (1, 1) sin(ϕ2 (ZT _r2 , t )) + TM (1, 2) cos(ϕ2 (ZT _r2 , t ))] sin(k2 (ZT _r2 )) − TM (1, 3) cos(k2 (ZT _r2 ))}τ2_1 (t )/τˆ
i_d , 0)), Ct3DT = min(max(Ct
ϕ0 (t )f /(ωτˆ )
= TM
(16)
H2 (ZT _r1 , τ (t )) = D cos(k2 (ZT _r2 )) − C sin(k2 (ZT _r2 )) = 0.
In comparison, point N can also be defined as the position of tooth 2 at time t for a tool spherical radius of RS2 − Ct2_1 and the equation is given by Eq. (12).
D = Ct2_1 sin(k2 (ZT _r2 )) = A sin(ϕ2 (ZT _r2 , t )) + B cos(ϕ2 (ZT _r2 , t )) = RS2 sin(k2 (ZT _r2 )) − RS1 sin(k1 (ZT _r1 )) cos(ωτ2_1 (t ) − ωτˆ + θ2_1 (ZT _r2 )) + f [TM (1, 1) sin(ϕ2 (ZT _r2 , t )) + TM (1, 2) cos(ϕ2 (ZT _r2 , t ))]τ2_1 (t )/τˆ
(12)
+ sin(ki−d (ZT _r (i−d) )) sin(ki (ZT _ri )) cos(ωτi_d (t ) − dωτˆ + θi_d (ZT _ri ))] + f {[TM (1, 1) sin(ϕi (ZT _ri , t )) + TM (1, 2) cos(ϕi (ZT _ri , t ))] sin(ki (ZT _ri )) − TM (1, 3) cos(ki (ZT _ri ))}τi_d (t )/τˆ .
(21)
2.2. Chip thickness model between adjacent flutes According to XT _N1 = XT _N2 , YT _N1 = YT _N2 and ZT _N1 = ZT _N2 , three collections of chip thickness can be derived from Eqs. (13)– (15), respectively. A = Ct2_1 sin(ϕ2 (ZT _r2 , t )) sin(k2 (ZT _r2 )) = RS2 sin(ϕ2 (ZT _r2 , t )) sin(k2 (ZT _r2 )) − RS1 sin(ϕ1 (ZT _r1 , t − τ2_1 (t ))) sin(k1 (ZT _r1 )) + fTM (1, 1)τ2_1 (t )/τˆ
(13)
2.3. Chip thickness models around engagement boundaries The previous machined surface is formed by the envelope of a series of cutters along the previous feed path. Since the feed dimension is always much smaller than that of the cutter, the envelope can be locally defined as a part of a cylinder with its axis parallel with the current feed direction as shown in Fig. 6. Crossfeed (cf ) and axial cutting depth (ap) are also defined in Fig. 6. The
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ϕ0 f /(ωτˆ ) 0 0
=
(R − CtBP ) sin k2 sin ϕ2 (R − CtBP ) sin k2 cos ϕ2 −(R − CtBP ) cos k2
+ TM
(23)
The position of the corresponding cutting tooth point MP is written as Eq. (24).
X wMP Y wMP Z wMP
ϕ0 f /(ωτˆ )
=
0 0
R sin k2 sin ϕ2 R sin k2 cos ϕ2 −R cos k2
+ TM
(24)
when substituting Eq. (23) to Eq. (22), a quadratic equation of local BP can be given by Eq. (25). chip thickness Ct 2
BP + H0 = 0 BP + H1 Ct H2 Ct
(25)
where 2 2 H2 = (Z wMP + Y wMP )/R2 2 2 H1 = −[2(Y wMP + cf )Y wMP + 2Z wMP ]/R 2 H0 = (Y wMP + cf )2 − R2 + Z wMP .
According to the fact that either axial cutting depth or cross-feed is much smaller than cutter radius in size in the finishing process, a constraint condition Q is given by Eq. (26) in detail to calculate the true root directly.
BP + H1 /2 Q = H2 ∗ Ct 2 2 BP − R) − R cfY wMP ]R2 = [(Z wMP + YWMP )(Ct 2 2 BP − R)(Z wMP ≤ (Ct + Y wMP + cfY wMP )/R2 2 BP − R)[Z wMP ≤ (Ct − (cf /22 )]R2
Fig. 4. (a) Chip thickness between adjacent teeth and (b) distribution of iterating cycles to solve the two constrains with respect to revolution angle from −90° to 270° and relative axial elevation from 0 to −1.
BP − R)[(R − ap)2 − (cf /22 )2 ]/R2 < 0. ≤ (Ct
(26)
The corresponding chip thickness induced by the previously ma BP = chined surface can be calculated using Eq. (27), where Ct 1
−H1 − (H12 − 4H2 H0 ) 2 /(2H2 ).
B P , 0). CtBP = max(Ct
(27)
An uncut surface can be simply expressed as a local plane parallel with the local workpiece surface at the corresponding cutter contact point as shown in Fig. 4(b). In coordinate system OXN YN ZN , the position of point NU on the plane can be given by Eq. (28). Z wN2 = ap − R
(28)
with the similar method used in CtBP , the chip thickness induced by uncut surface can be calculated analytically using Eq. (29). CtBU = max(0, R(ap − R − Z wMU ).Z wMU ).
(29)
In combination with all the chip thickness expressions given above, the final proposed model for chip thickness can be updated using Eq. (30). Cti (t , Z ) = min(Ct3DT , CtBP , CtBU ). 3. Verification and discussion
Fig. 5. Flow chart of 3D3T chip thickness model.
position of point NP on the previously machined surface can be given by Eq. (22) in the local workpiece coordinate system 2 = R2 . (Y wN2 + cf )2 + Z wN2
(22)
Meanwhile, with the definition of chip thickness Ct BP , the position of point NP can also be defined as a point on the cutting tooth with a tool spherical radius of R − CtBP as detailed in Eq. (23) X wN2 Y wN2 Z wN2
X wc2 Y wc2 Z wc2
=
X wr2 + Y wr2 Z wr2
(30)
3.1. Verification method and residual material In order to verify the proposed chip thicknesses model, a commercial software package (Unigraphics) was used to construct true envelopes of cutting tooth along a 3D3T guideline and to create 3D chip model. An HPA model (H) was also created in the software by modeling the cutter as a sphere since it is mostly adopted by the Z -map method with widely accessible accuracy in chip volume computation. Then volumes of the created 3D chip were directly measured (M) in software as benchmarks to assess
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Fig. 6. Engagement geometry for ball-end mill. (a) Chip thickness around previous machined boundary; (b) Chip thickness around uncut boundary; (c) Chip volume and its equivalent; (d) Area of section A. Table 1 Process parameters in static chip volume verifications with commercial software package (Unigraphics). Parameter and unit
Value
Feed per tooth (mm) Cross feed (mm) Axial cutting depth (mm) Lead angle (deg) Tilt angle (deg)
0.2 ±0.2 0.2 15 −15∼15
the accuracy of the proposed method (P). Process parameters and tool–workpiece inclinations involved are listed in Table 1. Both the prediction volumes of left-hand milling mode (L) and righthand milling mode (R) are examined for each model respectively as shown in Fig. 7, depending on whether the direction of crossfeed is positive or negative. When there is no tool inclination, the L mode is identical to the definition of up milling and the R mode is equivalent to down milling. In order to describe the volume of residual material within a tooth period, an ideally cut off volume per tooth without any residual material-between-teeth was also depicted in Fig. 7 as an approximate measurement (AM) of chip volume in finishing milling. It is equivalent to the volume that is extruded by length of the feed per tooth f from its orthogonal section A as shown in Fig. 4(c). The volume is given by Eq. (31), where SA is the exact area of section A. Vf = f × SA .
(31)
The predicted chip volume based on the proposed chip thickness method can be obtained by integrating in the spherical coordinate system and is given by Eq. (32). When the HPA or SPA method is applied, the cutter feed effect can be simplified and chip volume can be given by Eq. (33).
∫
z
VCt = −R
∫
φ2 φ1
Ct R sin kdφ +
φ τˆ
f (TM (1, 1) cos φ − TM (1, 2) sin φ)
dz sin k
(32)
Fig. 7. Chip volume comparisons with direct measurements in commercial software package (Unigraphics). PL/PR—Predicted volume for Left/Right-hand mode; ML/MR—Measured volume for Left/Right-hand mode; MH—Measured volume for HPA (hemisphere paths assumption) model; AM— chip volume of Approximate Measurement method.
∫
z
VCt = −R
∫
φ2 φ1
Ct [R sin kdφ]
dz sin k
.
(33)
Comparing the measured values ML, MR and MH, as well as AM in Fig. 7 the differences are near-negligible in terms of the chip volume values; although the chip volumes of the 3D3T model are a bit smaller than that of the HPA model either in the L mode or in the R mode. However, the difference of measured chip volume between model 3D3T and AM, namely the volume of residual material, is at least three times larger than that of the HPA model, as the volume of residual material is only about several percents of chip volume. It means that more prediction accuracy can be obtained with the 3D3T model when describing machined surface profile, which is sometimes equivalent to the volume of residual material. Comparing the predicted chip volume PL and PR with the measured ones, it can be found that the proposed chip thickness
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Table 2 Model error comparisons for different cutting depths at T = 0°, L = 10°, f = 0.1 mm, R = 5 mm. Test num.
Adoc (mm)
pf (mm)
ERFL (%)
ERFR (%)
SPAL (%)
SPAR (%)
HPAL (%)
HPAR (%)
PL (%)
PR (%)
1 2 3 4 5 6 7
0.4 0.4 0.4 0.2 0.2 0.2 0.2
0.6 0.4 0.2 0.6 0.4 0.2 Slot
17 23 41 19 26 44 19
19 26 44 21 28 48
−52 −53 −55 −53 −54 −56 −14
−52 −53 −55 −53 −54 −56
−0.213 −0.123 −0.045 −0.115 −0.031
−0.213 −0.123 −0.045 −0.115 −0.031
−0.211 −0.128
−0.214 −0.129
0.175 1.55
0.175
L R
0.042
0.049
−0.116
−0.115 −0.030
0.032 0.167 0.38
0.183
Left-hand feed mode. Right-hand feed mode.
Fig. 8. Static chip geometries and volume errors in tool inclination and cutter runout. (a1) Static chip thicknesses and tool contact regions; (a2) Chip volume errors with various tool inclinations; (b1) Chip thickness contour in cutter runout with e = 0.001 mm, ρ = π/6; (b2) Chip volume errors in cutter runout.
method has an acceptable accuracy as the same level of HPA method for chip volume computation, since both the predicted and the measured values are quite close. Besides, the proposed method can result in better agreement for residual material and profile prediction. 3.2. Model comparisons in static chip volume As the discussed in the previous section, the volume of residual material is much smaller than the static chip volume in finishing milling. Instead of direct measuring in software, the approximate measurement (AM) method is used here to assess the accuracy of chip thickness models under a wider range of process parameters. First, the integrated chip volumes VCt of different chip thickness models are obtained by Eq. (32) or Eq. (33), then compared with the ideally cut off volume Vf in Eq. (31). The closer the volume VCt to Vf , the more accurate is the chip thickness model. The prediction error can be approximately given by Eq. (34). err = (VCt − Vf )/Vf × 100%.
(34)
According to Eq. (34), accuracy comparison tests of the proposed model and the existing models SPA, HPA as well as ERF have been carried out with various parameters such as cutting depth, tool inclination and cutter runout. Specifically, models SPA and HPA are also combined with the proposed engagement boundary model of chip thickness to examine the accuracy improvement by model 3D3T Model ERF is constructed by filtering model 3D3T with the engagement region. As shown in Table 2, the accuracy of chip volume of model HPA is still of the same level with that of the proposed model in a wide range of cutting depth, which indicates that HPA has little effect on static chip volume. As far as model SPA is concerned, a quadratic growth comes out for the underestimated error induced by the product of two sine functions, which is even over 50% for some cases in Table 2. As the chip thicknesses of model ERF are much steeper around the engagement boundaries, the overestimated error of prediction volume can be greater than 40%. However, the errors for the proposed combination model are always within 5%. In addition, the volume accuracy of the proposed model can
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Fig. 9. Dynamics comparisons between the proposed model and the HPA model. (a) Tooth delays; (b) Relative dynamic chip thickness; (c) SLD comparisons with experimental results in literature [17]; and (d) Shift effects of SLD due to time-varying delays. Table 3 Parameters used in dynamic discussions [17]. Modal 1
Modal 2
Modal 3
Fn = 1826.7 Hz ζ = 9.993e − 3 M = 0.0418 Kg
Fn = 1934.4 Hz ζ = 9.584e − 3 M = 0.0824 Kg
Fn = 2028.4 Hz ζ = 9.726e − 3 M = 0.0852 Kg
be illustrated again for the case of tool inclination and cutter runout as shown in Fig. 8, although the cutter contact regions and instantaneous chip thicknesses are quite different from each other. 3.3. Model comparisons in milling dynamics Besides the static chip volume, differences are also emerged in time-varying delays τi_1 (t ). Unlike the HPA model, the delays are no longer constant like Tˆ for the 3D3T model as shown in Fig. 9(a) for the 7th process parameter in Table 2. When the revolution angle is small in the L mode, the delays are shrunk as well as the corresponding chip thicknesses. On the contrary, the delays and chip thicknesses are enlarged for the R mode as the revolution angle is close to π . Both the earlier effect in L and the later effect in R can be almost enhanced to 10% when the cutting edge approaches the cutter tip. As a result, the average D2S, defined as a ratio of dynamic chip thickness to the corresponding static one, appears more different between using the proposed model and the HPA model in light-cut finishing. It is distinguished from the consistent performance in slot milling as shown in Fig. 9(b). The semi-discretization method has proved capable of determining system stabilities for a nonautonomous system in either a
Process parameters
Cutting force coefficients from orthogonal database
R = 4 mm
V m/min
l0 = 30 deg L = T = 15 deg f = 0.05 mm cf = 0.1 mm
Ct mm r = 0.40 + 0.0005V + 0.6Ct β = 26.8 − 0.0313V + 11.77Ct deg Tshear = 450.3 + 0.4V + 227.5Ct MPa
constant delay [19] or a periodic delay [14,20]. With this method, stabilities of light-cut finishing have been assessed using the existing model HPA and the proposed model. Dynamic properties of tool force coefficients as well as process parameters adopted in literature [17] were used here to examine the proposed chip thickness model, as listed in Table 3. The proposed chip thickness model can be validated by comparing the predicted stability lobes diagram (SLD) with that of experiments in literature [17], as depicted in Fig. 9(c). As the average time-varying delay is only 0.34% sooner than the pitch of the tooth, the differences against that of model HPA can be ignored. Another set of parameters, including f = 0.2 mm, cf = −0.2 mm, L = 15°, T = 0°, was carried out to show the differences between the proposed model and HPA, since a larger later delay can be captured. Although the average later effect of the delays is only about 2%, the right-shift effects of SLD have been observed for the proposed model as depicted in Fig. 9(d), which can draw a bigger different stability region for machining process optimization. This is because the locations of peak of SLD are defined by the ratio between the dominant natural frequency of the machine-tool system and the tooth passing frequency; and either the earlier effect or the later effect of time-varying delays
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can be amplified as the ratio changes. Therefore, the differences in dynamic performance between using the proposed model and the HPA model seem to be much more significant in comparison with that in static chip thickness modeling. 4. Conclusions In this paper, an in-depth geometric model is established to determine instantaneous chip thickness for a 5-axis ball-end finish milling. Applying the 3D3T expression of cutting edge, a numerical method which is able to handle cases with various feedrate, tool–workpiece inclination and cutter runout, is proposed. Then chip thickness around engagement boundaries is modeled directly to envisage the amplified overestimation effect in the finish process. In combination with the direct measuring chip volume of a commercial software package (Unigraphics), it is shown that one order reduction can be achieved for both the underestimated error in classical model SPA and the overestimated error in the widely used method ERF. Dynamic properties including timevarying delays, dynamic chip thicknesses and SLD are generated to reveal that the proposed model is also of a different nature compared with the existing model HPA. As the linear feed assumption is used for most conditions, the cutter inclinations for the adjacent two flutes are considered the same in the proposed chip thickness model. The relationship between previous cutting surface and current cutting tooth need to be modified for the case of 5-axis flow cut finishing or hole milling where component features are of the same scale with cutter dimensions, and time-varying delays might become more distinguished. In future work, modifications of local gesture transformation will be considered to describe the milling stabilities in 5-axis flow cut finishing. Acknowledgment This work was supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 50821003).
References [1] Lee P, Altinta Y. Prediction of ball-end milling forces from orthogonal cutting data. Int J Mach Tools Manuf 1996;36(9):1059–72. [2] Engin S, Altintas Y. Mechanics and dynamics of general milling cutters: Part I: Helical end mills. Int J Mach Tools Manuf 2001;41(15):2195. [3] Wang JJ, Zheng CM. Identification of shearing and ploughing cutting constants from average forces in ball-end milling. Int J Mach Tools Manuf 2002;42(6): 695–705. [4] Gradiek J, Kalveram M, Weinert K. Mechanistic identification of specific force coefficients for a general end mill. Int J Mach Tools Manuf 2004;44(4): 401–14. [5] Ozturk B, Lazoglu I, Erdim H. Machining of free-form surfaces. Part II: Calibration and forces. Int J Mach Tools Manuf 2006;46(7–8):736–46. [6] Ferry WB, Altintas Y. Virtual five-axis flank milling of jet engine impellers— part I: mechanics of five-axis flank milling. J Manuf Sci Eng 2008;130(1): 011005. [7] Feng HY, Ning SU. A mechanistic cutting force model for 3D ball-end milling. J Manuf Sci Eng 2001;123(1):23–9. [8] Altintas Y, Lee P. Mechanics and dynamics of ball end milling. J Manuf Sci Eng 1998;120(4):684–92. [9] Zhu R, Kapoor SG, DeVor RE. Mechanistic modeling of the ball end milling process for multi-axis machining of free-form surfaces. J Manuf Sci Eng 2001; 123(3):369. [10] Sun Y, et al. Estimation and experimental validation of cutting forces in ballend milling of sculptured surfaces. Int J Mach Tools Manuf 2009;49(15): 1238–44. [11] Roth D, Ismail F, Bedi S. Mechanistic modelling of the milling process using an adaptive depth buffer. Comput Aided Des 2003;35(14):1287. [12] Khachan S, Ismail F. Machining chatter simulation in multi-axis milling using graphical method. Int J Mach Tools Manuf 2009;49(2):163–70. [13] Li HZ, Liu K, Li XP. A new method for determining the undeformed chip thickness in milling. J Mater Process Technol 2001;113(1–3):378–84. [14] Faassen RPH, et al. An improved tool path model including periodic delay for chatter prediction in milling. J Comput Nonlinear Dyn 2007;2(2):167. [15] Ozturk B, Lazoglu I. Machining of free-form surfaces. Part I: analytical chip load. Int J Mach Tools Manuf 2006;46(7–8):728–35. [16] Shamoto E, Akazawa K. Analytical prediction of chatter stability in ball end milling with tool inclination. CIRP J Manuf Sci Technol 2009;58(1):351–4. [17] Ozturk E, Budak E. Dynamics and stability of five-axis ball-end milling. J Manuf Sci Eng 2010;132:021003. [18] Ozturk E, Budak E. Modeling of 5-axis milling processes. Mach Sci Technol 2007;11(3):287–311. [19] Insperger T, Stépán G. Semi-discretization method for delayed systems. Internat J Numer Methods Engrg 2002;55(5):503–18. [20] Insperger T, Stépán G. Updated semi-discretization method for periodic delaydifferential equations with discrete delay. Internat J Numer Methods Engrg 2004;61(1):117–41.
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Computer-Aided Design journal homepage: www.elsevier.com/locate/cad
An accuracy algorithm for chip thickness modeling in 5-axis ball-end finish milling Xin-Guang Liang a,∗ , Zhen-Qiang Yao a,b a
School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China
b
State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, PR China
article
info
Article history: Received 26 October 2010 Accepted 25 April 2011 Keywords: Chip thickness 5-axis mill Ball-end finishing 3D trochoidal tooth trajectory Engagement boundary
abstract Chip thickness in milling is one of the most fundamental parameters, which can significantly affect cutting force, cutting heat, cutting stability and machined surface topography for computer-aided process planning. In this paper, a combination of a three-dimensional trochoidal tooth trajectory model (3D3T) and engagement-boundary chip model is developed to determine instantaneous chip thickness in 5-axis ball-end finish milling. In comparison with the chip volume measured in a commercial software package (Unigraphics) the accuracy of the proposed model has been numerically validated with various process parameters including cutting depth, tool–workpiece inclination and cutter runout. The differences in time-varying delay and dynamic chip thickness as well as stability are compared with different models to show the impact of using 3D3T mechanism for chip thickness modeling in 5-axis ball-end finish. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction 5-axis ball-end milling of 3D free-form surfaces such as dies, molds, turbines and various aerospace components is gaining wider applications. Chip thickness, one of the most fundamental parameters in milling process planning, can significantly affect cutting force, cutting heat, cutting stability and machined surface topography. Over the past 20 years, chip thickness modeling in 5-axis ball-end milling has been receiving great attention. In general, most chip thickness models were more or less dependent on two basic assumptions. One is the hemisphere paths assumption (HPA) that trajectories of cutting tooth are decomposed into pure translation of tool axis and revolution around the tool axis in series, and spheric rules can be transplanted expediently. The other can be regarded as the sine product assumption (SPA) in which the chip thickness model is simplified as Ct = f sin ϕ sin k, namely, feed per tooth f , revolution angle ϕ and immersion angle k become three decoupled parameters. Both assumptions can make it more convenient and efficient to analytically model chip geometry, cutting force and stability in ball-end milling [1–7]. Unfortunately, with the development of research both HPA- and SPA-induced model errors have been detected either in axial or in circumferential, especially for lightcut finishing where cutting depth, width and feed are all within 10% of cutter radius.
∗
Corresponding author. Tel.: +86 21 3420 6583. E-mail addresses: [email protected], [email protected] (X.-G. Liang). 0010-4485/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cad.2011.04.012
Many efforts have been involved in using the methods of 3D grid descriptions or constructive solid geometry (CSG) and indeed improved modeling accuracy of chip thickness. Altintas and Lee [8] proposed a Z -map method to eliminate the circumferential error of sin ϕ in SPA. Zhu et al. [9] proposed a distance method which needs to calculate the intersection of the previous tooth envelope surface and a vector line passing through the instantaneous ball center. Although this model can remove the whole error induced by SPA, the intersection point would be much more difficult to obtain directly without HPA. Sun et al. [10] proposed a 2D trochoidal tooth paths method to calculate the relative chip thickness in 3-axis ballend milling, which can partly get rid of HPA. However, it was not mentioned how to deal with the inevitable axial error of SPA when determining the definite chip thickness. Roth et al. [11] proposed an adaptive depth buffer method to save computing memory by sizing the depth buffer to the tool as opposed to the workpiece. Khachan and Ismail [12] furthered this solid graphics method by describing the chip thickness in the form of chip volume, and simulated machining chatter in 5-axis milling. This method can be extended to the true cutting path as trochoid and thoroughly eliminate chip thickness errors induced by SPA and HPA. However, it did not clearly describe how to determine the non-uniform delays between adjacent teeth. Fixing the uniform delays to the period of tooth can cause different cutting force and dynamics. Li et al. [13] applied Taylor’s series to express the non-uniform delays and revealed the errors of traditional model in 3-axis milling. Faassen et al. [14] discussed the influences of non-uniform delays on the shift of stability lobes in 3-axis milling of low radial immersion.
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Besides the two assumptions, engagement region filter (ERF) is another simplified method commonly used to determine the activity of chip thickness. As long as tooth segments locate in engagement region, the corresponding chip thicknesses would be activated by multiplying one, or else they will be replaced by zeros [6,15–18]. However, ERF can result in discontinuous steps of chip thickness around engagement boundaries, making chip thickness always overestimated. The overestimated chip thickness can also become more significant in finish milling. Until now, there have been few existing models that can completely step out of the two assumptions and ERF simplification, especially in 5-axis milling. This paper presents a three-dimensional trochoidal tooth trajectory (3D3T) model to describe the definite chip thickness for 5-axis ball-end milling in finishing processes without using the assumptions HPA and SPA. An engagement boundary chip model is also established to eliminate the overestimated chip thickness error efficiently. Based on the fact that chip volume can directly reflect the accuracy of chip thickness [12], numerical chip volume generated from the proposed model and a commercial software package are compared to validate the accuracy of the proposed model by studying the effects of cutting depth, tool inclination and cutter runout. Impacts of the proposed model on milling stability and stable vibration cycle are mentioned for future applications.
Fig. 1. Cutter geometries and runout definitions.
2. Chip thickness modeling with 3D3T 2.1. Cutter runout and 3D3T in 5-axis milling A schematic representation of cutter geometry and its runout is shown in Fig. 1. Note that R denotes the radius of the cutter and l0 represents helix angle. Point P located on the ith flute, has elevation ZT , revolution radius r and lag angle Ψ . The lag angle Ψ is given by Eq. (1).
ψ(ZT ) = (R + ZT ) tan l0 /R.
Due to the presence of cutter runout, the tool coordinate system OXT YT ZT and the spindle coordinate system OS XS YS ZS no longer coincide with each other; their relative positions can be described by runout parameters e and ρ as shown in Fig. 1. Cutting radius, Rti ; spherical radius, RSi ; immersion angles, ki (ZT ) as well as cutter runout induced lag angle βi (ZT ) can be updated in the spindle coordinate system OS XS YS ZS using Eqs. (2)–(5), respectively where Yi (ZT ) = ρ + Ψ (ZT ) + 2π (i − 1)/n. Rti = RSi =
ri (ZT )2 + e2 − 2ri (ZT )e cos γi (ZT )
Rti2 + ZT2
ki (ZT ) = cos
−1
(ZT /RSi )
βi (ZT ) = [sign(sin γi )
sign(sin γi )
Fig. 2. Cutting orientations in 5-axis milling.
(1)
] cos [(
Rti2
1 0 0
0 cos(t ) sin(t )
0 − sin(t ) cos(t )
cos(l′ ) 0 − sin(l′ )
0 1 0
sin(l′ ) 0 . cos(l′ )
(2)
TM =
(3)
Therefore, parameters of tooth trajectory in OXN YN ZN and process parameters in OS XS YS ZS can be transformed from Eq. (10). Two 3D trochoidal tooth trajectories, corresponding to two distinguished points on the same cutting edge are depicted in Fig. 3.
(4) −1
In the 5-axis ball-end milling case, the tool axis is no more orthogonal to the feed direction due to the tool–workpiece inclination. For an arbitrary linear feed portion of the whole tool trajectory, the involved coordinate systems are depicted in Fig. 2. A transformation from OS XS YS ZS to OXN YN ZN , denoted by TM , can be derived from Eq. (9) where l and t denote lead angle and tilt angle respectively, and l′ = tan−1 (tan(l) cos(t )).
+ ri − e )/(2Rti ri )]. (5) 2
(9)
Combining the influence of cutter runout and flute helix, the total lag angle δi (ZT ), which can be used to determine the actual revolution angle of the cutting edge, is calculated using Eq. (6).
δi (ZT ) = βi (ZT ) + ψ(ZT ).
In 3D3T modeling, instantaneous uncut chip thickness can be calculated through the instantaneous position of the cutter center as depicted in Fig. 3. Point M is the instantaneous location on the ith cutting edge of ZT , point N denotes the point on the dth previous cutting edge; point Oi and point Oi_d represent their corresponding ball centers respectively. Additionally, line Oi M just passes by point N. τi_d (t ) is defined as the delay that is involved when calculating the chip thickness that tooth i removes at time t with respect to the tooth i − d. For example as shown in Fig. 3, τ2_1 (t ) is the difference between the time when tooth 2 is at point M and
(6)
Then, the revolution angle ϕi (ZT , t ) can be given by Eq. (7).
ϕi (ZT , t ) = ϕi (−R, 0) + ωt − δi (ZT ).
(7)
The flutes interval angle for a cutter of n flutes can be expressed by Eq. (8), where ω and τˆ are defined as spindle angular speed and pitch of tooth respectively.
ωτˆ = ϕi−1 (−R, 0) − ϕi (−R, 0) = 2π /n.
(8)
XN YN ZN
XT YT ZT
= TM
XT YT ZT
or
XN YN ZN
′
= TM
.
(10)
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B = Ct2_1 cos(ϕ2 (ZT _r2 , t )) sin(k2 (ZT _r2 )) = RS2 cos(ϕ2 (ZT _r2 , t )) sin(k2 (ZT _r2 )) − RS1 cos(ϕ1 (ZT _r1 , t − τ2_1 (t ))) sin(k1 (ZT _r1 )) + fTM (1, 2)τ2_1 (t )/τˆ
(14)
C = Ct2_1 cos(k2 (ZT _r2 )) = RS2 cos(k2 (ZT _r2 )) − RS1 cos(k1 (ZT _r1 )) − fTM (1, 3)τ2_1 (t )/τˆ .
(15)
The local geometry parameters of point M at time t in the above three equations, including k2 , ϕ2 , ZT _r2 and RS2 , are easy to express as input parameters. Each equation can directly formulate the chip thickness. However, it will result in an error when the denominator of any of these equations approaches zero. Therefore, the Pythagorean proposition is applied twice to eliminate the parameter ϕ2 and k2 on the left, yielding collection D in Eq. (16) and instantaneous chip thickness Ct in Eq. (17) which can always be validated whatever ϕ2 or k2 is getting zero or not.
Fig. 3. (a) 3D3Ts and (b) previous cutting surface descriptions.
the time when tooth 1 is at point N. Because feed per tooth is always rather smaller compared with the dimension of the sculpture feature on the workpiece surface, linear feed assumption can be adopted within the scale of feed per tooth. The position of point N on tooth 1 can be described as a sum of the position of the instantaneous ball center O1 and the position of cutting edge relative to O1 at time t − τ2_1 (t ). The expression is given by Eq. (11). XT _N1 YT _N1 ZT _N1
XT _c1 YT _c1 ZT _c1
=
XT _r1 + YT _r1 ZT _r1
ϕ0 (t − τ2_1 (t ))f /(ωτˆ )
′
0 0
= TM +
RS1 sin(k1 (ZT _r1 )) sin(ϕ1 (ZT _r1 , t − τ2_1 (t ))) RS1 sin(k1 (ZT _r1 )) cos(ϕ1 (ZT _r1 , t − τ2_1 (t ))) . RS1 cos(k1 (ZT _r1 ))
(11)
XT _N2 YT _N2 ZT _N2 ′
XT _c2 YT _c2 ZT _c2
=
XT _r2 + YT _r2 ZT _r2
(17)
where θi_d (ZT _ri ) = δi−d (ZT _r (i−d) ) − δi (ZT _ri ). The corresponding delay τ2_1 and previous tooth elevation ZT _r1 can be solved firstly by either numerically iterating Eqs. (18) and (19) for an exact solution or linearizing them for an approximate solution. When initial conditions τ2_1 = 0 and ZT _r1 = ZT _r2 are used for iteration with the Newton–Raphson method, exact (99.9% approximate) solutions can be obtained within two or three iterations for most of the points on the previous machined surface as shown in Fig. 4. H1 (ZT _r1 , τ (t )) = A cos(ϕ2 (ZT _r2 , t )) − B sin(ϕ2 (ZT _r2 , t )) = 0 (18) (19)
Then, the 3D3T-based instantaneous chip thickness for an arbitrary tooth i can be numerically derived from Eq. (20). The algorithm can also be explained in Fig. 5. d = 1, 2, . . . , n (20) where Ct i_d denotes the static chip thickness between arbitrary tooth i and tooth i − d, as expressed in Eq. (21). i_d = RSi − RSi−d [cos(ki−d (ZT _r (i−d) )) cos(ki (ZT _ri )) Ct
0 0
(RS2 − Ct2_1 ) sin(k2 (ZT _r2 )) sin(ϕ2 (ZT _r2 , t )) + (RS2 − Ct2_1 ) sin(k2 (ZT _r2 )) cos(ϕ2 (ZT _r2 , t )) . −(RS2 − Ct2_1 ) cos(k2 (ZT _r2 ))
Ct2_1 = C cos(k2 (ZT _r2 )) + D sin(k2 (ZT _r2 )) = RS2 − RS1 [cos(k1 (ZT _r1 )) cos(k2 (ZT _r2 )) + sin(k1 (ZT _r1 )) sin k2 (ZT _r2 ) cos(ωτ2_1 (t ) − ωτˆ + θ2_1 (ZT _r2 ))] + f {[TM (1, 1) sin(ϕ2 (ZT _r2 , t )) + TM (1, 2) cos(ϕ2 (ZT _r2 , t ))] sin(k2 (ZT _r2 )) − TM (1, 3) cos(k2 (ZT _r2 ))}τ2_1 (t )/τˆ
i_d , 0)), Ct3DT = min(max(Ct
ϕ0 (t )f /(ωτˆ )
= TM
(16)
H2 (ZT _r1 , τ (t )) = D cos(k2 (ZT _r2 )) − C sin(k2 (ZT _r2 )) = 0.
In comparison, point N can also be defined as the position of tooth 2 at time t for a tool spherical radius of RS2 − Ct2_1 and the equation is given by Eq. (12).
D = Ct2_1 sin(k2 (ZT _r2 )) = A sin(ϕ2 (ZT _r2 , t )) + B cos(ϕ2 (ZT _r2 , t )) = RS2 sin(k2 (ZT _r2 )) − RS1 sin(k1 (ZT _r1 )) cos(ωτ2_1 (t ) − ωτˆ + θ2_1 (ZT _r2 )) + f [TM (1, 1) sin(ϕ2 (ZT _r2 , t )) + TM (1, 2) cos(ϕ2 (ZT _r2 , t ))]τ2_1 (t )/τˆ
(12)
+ sin(ki−d (ZT _r (i−d) )) sin(ki (ZT _ri )) cos(ωτi_d (t ) − dωτˆ + θi_d (ZT _ri ))] + f {[TM (1, 1) sin(ϕi (ZT _ri , t )) + TM (1, 2) cos(ϕi (ZT _ri , t ))] sin(ki (ZT _ri )) − TM (1, 3) cos(ki (ZT _ri ))}τi_d (t )/τˆ .
(21)
2.2. Chip thickness model between adjacent flutes According to XT _N1 = XT _N2 , YT _N1 = YT _N2 and ZT _N1 = ZT _N2 , three collections of chip thickness can be derived from Eqs. (13)– (15), respectively. A = Ct2_1 sin(ϕ2 (ZT _r2 , t )) sin(k2 (ZT _r2 )) = RS2 sin(ϕ2 (ZT _r2 , t )) sin(k2 (ZT _r2 )) − RS1 sin(ϕ1 (ZT _r1 , t − τ2_1 (t ))) sin(k1 (ZT _r1 )) + fTM (1, 1)τ2_1 (t )/τˆ
(13)
2.3. Chip thickness models around engagement boundaries The previous machined surface is formed by the envelope of a series of cutters along the previous feed path. Since the feed dimension is always much smaller than that of the cutter, the envelope can be locally defined as a part of a cylinder with its axis parallel with the current feed direction as shown in Fig. 6. Crossfeed (cf ) and axial cutting depth (ap) are also defined in Fig. 6. The
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ϕ0 f /(ωτˆ ) 0 0
=
(R − CtBP ) sin k2 sin ϕ2 (R − CtBP ) sin k2 cos ϕ2 −(R − CtBP ) cos k2
+ TM
(23)
The position of the corresponding cutting tooth point MP is written as Eq. (24).
X wMP Y wMP Z wMP
ϕ0 f /(ωτˆ )
=
0 0
R sin k2 sin ϕ2 R sin k2 cos ϕ2 −R cos k2
+ TM
(24)
when substituting Eq. (23) to Eq. (22), a quadratic equation of local BP can be given by Eq. (25). chip thickness Ct 2
BP + H0 = 0 BP + H1 Ct H2 Ct
(25)
where 2 2 H2 = (Z wMP + Y wMP )/R2 2 2 H1 = −[2(Y wMP + cf )Y wMP + 2Z wMP ]/R 2 H0 = (Y wMP + cf )2 − R2 + Z wMP .
According to the fact that either axial cutting depth or cross-feed is much smaller than cutter radius in size in the finishing process, a constraint condition Q is given by Eq. (26) in detail to calculate the true root directly.
BP + H1 /2 Q = H2 ∗ Ct 2 2 BP − R) − R cfY wMP ]R2 = [(Z wMP + YWMP )(Ct 2 2 BP − R)(Z wMP ≤ (Ct + Y wMP + cfY wMP )/R2 2 BP − R)[Z wMP ≤ (Ct − (cf /22 )]R2
Fig. 4. (a) Chip thickness between adjacent teeth and (b) distribution of iterating cycles to solve the two constrains with respect to revolution angle from −90° to 270° and relative axial elevation from 0 to −1.
BP − R)[(R − ap)2 − (cf /22 )2 ]/R2 < 0. ≤ (Ct
(26)
The corresponding chip thickness induced by the previously ma BP = chined surface can be calculated using Eq. (27), where Ct 1
−H1 − (H12 − 4H2 H0 ) 2 /(2H2 ).
B P , 0). CtBP = max(Ct
(27)
An uncut surface can be simply expressed as a local plane parallel with the local workpiece surface at the corresponding cutter contact point as shown in Fig. 4(b). In coordinate system OXN YN ZN , the position of point NU on the plane can be given by Eq. (28). Z wN2 = ap − R
(28)
with the similar method used in CtBP , the chip thickness induced by uncut surface can be calculated analytically using Eq. (29). CtBU = max(0, R(ap − R − Z wMU ).Z wMU ).
(29)
In combination with all the chip thickness expressions given above, the final proposed model for chip thickness can be updated using Eq. (30). Cti (t , Z ) = min(Ct3DT , CtBP , CtBU ). 3. Verification and discussion
Fig. 5. Flow chart of 3D3T chip thickness model.
position of point NP on the previously machined surface can be given by Eq. (22) in the local workpiece coordinate system 2 = R2 . (Y wN2 + cf )2 + Z wN2
(22)
Meanwhile, with the definition of chip thickness Ct BP , the position of point NP can also be defined as a point on the cutting tooth with a tool spherical radius of R − CtBP as detailed in Eq. (23) X wN2 Y wN2 Z wN2
X wc2 Y wc2 Z wc2
=
X wr2 + Y wr2 Z wr2
(30)
3.1. Verification method and residual material In order to verify the proposed chip thicknesses model, a commercial software package (Unigraphics) was used to construct true envelopes of cutting tooth along a 3D3T guideline and to create 3D chip model. An HPA model (H) was also created in the software by modeling the cutter as a sphere since it is mostly adopted by the Z -map method with widely accessible accuracy in chip volume computation. Then volumes of the created 3D chip were directly measured (M) in software as benchmarks to assess
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Fig. 6. Engagement geometry for ball-end mill. (a) Chip thickness around previous machined boundary; (b) Chip thickness around uncut boundary; (c) Chip volume and its equivalent; (d) Area of section A. Table 1 Process parameters in static chip volume verifications with commercial software package (Unigraphics). Parameter and unit
Value
Feed per tooth (mm) Cross feed (mm) Axial cutting depth (mm) Lead angle (deg) Tilt angle (deg)
0.2 ±0.2 0.2 15 −15∼15
the accuracy of the proposed method (P). Process parameters and tool–workpiece inclinations involved are listed in Table 1. Both the prediction volumes of left-hand milling mode (L) and righthand milling mode (R) are examined for each model respectively as shown in Fig. 7, depending on whether the direction of crossfeed is positive or negative. When there is no tool inclination, the L mode is identical to the definition of up milling and the R mode is equivalent to down milling. In order to describe the volume of residual material within a tooth period, an ideally cut off volume per tooth without any residual material-between-teeth was also depicted in Fig. 7 as an approximate measurement (AM) of chip volume in finishing milling. It is equivalent to the volume that is extruded by length of the feed per tooth f from its orthogonal section A as shown in Fig. 4(c). The volume is given by Eq. (31), where SA is the exact area of section A. Vf = f × SA .
(31)
The predicted chip volume based on the proposed chip thickness method can be obtained by integrating in the spherical coordinate system and is given by Eq. (32). When the HPA or SPA method is applied, the cutter feed effect can be simplified and chip volume can be given by Eq. (33).
∫
z
VCt = −R
∫
φ2 φ1
Ct R sin kdφ +
φ τˆ
f (TM (1, 1) cos φ − TM (1, 2) sin φ)
dz sin k
(32)
Fig. 7. Chip volume comparisons with direct measurements in commercial software package (Unigraphics). PL/PR—Predicted volume for Left/Right-hand mode; ML/MR—Measured volume for Left/Right-hand mode; MH—Measured volume for HPA (hemisphere paths assumption) model; AM— chip volume of Approximate Measurement method.
∫
z
VCt = −R
∫
φ2 φ1
Ct [R sin kdφ]
dz sin k
.
(33)
Comparing the measured values ML, MR and MH, as well as AM in Fig. 7 the differences are near-negligible in terms of the chip volume values; although the chip volumes of the 3D3T model are a bit smaller than that of the HPA model either in the L mode or in the R mode. However, the difference of measured chip volume between model 3D3T and AM, namely the volume of residual material, is at least three times larger than that of the HPA model, as the volume of residual material is only about several percents of chip volume. It means that more prediction accuracy can be obtained with the 3D3T model when describing machined surface profile, which is sometimes equivalent to the volume of residual material. Comparing the predicted chip volume PL and PR with the measured ones, it can be found that the proposed chip thickness
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Table 2 Model error comparisons for different cutting depths at T = 0°, L = 10°, f = 0.1 mm, R = 5 mm. Test num.
Adoc (mm)
pf (mm)
ERFL (%)
ERFR (%)
SPAL (%)
SPAR (%)
HPAL (%)
HPAR (%)
PL (%)
PR (%)
1 2 3 4 5 6 7
0.4 0.4 0.4 0.2 0.2 0.2 0.2
0.6 0.4 0.2 0.6 0.4 0.2 Slot
17 23 41 19 26 44 19
19 26 44 21 28 48
−52 −53 −55 −53 −54 −56 −14
−52 −53 −55 −53 −54 −56
−0.213 −0.123 −0.045 −0.115 −0.031
−0.213 −0.123 −0.045 −0.115 −0.031
−0.211 −0.128
−0.214 −0.129
0.175 1.55
0.175
L R
0.042
0.049
−0.116
−0.115 −0.030
0.032 0.167 0.38
0.183
Left-hand feed mode. Right-hand feed mode.
Fig. 8. Static chip geometries and volume errors in tool inclination and cutter runout. (a1) Static chip thicknesses and tool contact regions; (a2) Chip volume errors with various tool inclinations; (b1) Chip thickness contour in cutter runout with e = 0.001 mm, ρ = π/6; (b2) Chip volume errors in cutter runout.
method has an acceptable accuracy as the same level of HPA method for chip volume computation, since both the predicted and the measured values are quite close. Besides, the proposed method can result in better agreement for residual material and profile prediction. 3.2. Model comparisons in static chip volume As the discussed in the previous section, the volume of residual material is much smaller than the static chip volume in finishing milling. Instead of direct measuring in software, the approximate measurement (AM) method is used here to assess the accuracy of chip thickness models under a wider range of process parameters. First, the integrated chip volumes VCt of different chip thickness models are obtained by Eq. (32) or Eq. (33), then compared with the ideally cut off volume Vf in Eq. (31). The closer the volume VCt to Vf , the more accurate is the chip thickness model. The prediction error can be approximately given by Eq. (34). err = (VCt − Vf )/Vf × 100%.
(34)
According to Eq. (34), accuracy comparison tests of the proposed model and the existing models SPA, HPA as well as ERF have been carried out with various parameters such as cutting depth, tool inclination and cutter runout. Specifically, models SPA and HPA are also combined with the proposed engagement boundary model of chip thickness to examine the accuracy improvement by model 3D3T Model ERF is constructed by filtering model 3D3T with the engagement region. As shown in Table 2, the accuracy of chip volume of model HPA is still of the same level with that of the proposed model in a wide range of cutting depth, which indicates that HPA has little effect on static chip volume. As far as model SPA is concerned, a quadratic growth comes out for the underestimated error induced by the product of two sine functions, which is even over 50% for some cases in Table 2. As the chip thicknesses of model ERF are much steeper around the engagement boundaries, the overestimated error of prediction volume can be greater than 40%. However, the errors for the proposed combination model are always within 5%. In addition, the volume accuracy of the proposed model can
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Fig. 9. Dynamics comparisons between the proposed model and the HPA model. (a) Tooth delays; (b) Relative dynamic chip thickness; (c) SLD comparisons with experimental results in literature [17]; and (d) Shift effects of SLD due to time-varying delays. Table 3 Parameters used in dynamic discussions [17]. Modal 1
Modal 2
Modal 3
Fn = 1826.7 Hz ζ = 9.993e − 3 M = 0.0418 Kg
Fn = 1934.4 Hz ζ = 9.584e − 3 M = 0.0824 Kg
Fn = 2028.4 Hz ζ = 9.726e − 3 M = 0.0852 Kg
be illustrated again for the case of tool inclination and cutter runout as shown in Fig. 8, although the cutter contact regions and instantaneous chip thicknesses are quite different from each other. 3.3. Model comparisons in milling dynamics Besides the static chip volume, differences are also emerged in time-varying delays τi_1 (t ). Unlike the HPA model, the delays are no longer constant like Tˆ for the 3D3T model as shown in Fig. 9(a) for the 7th process parameter in Table 2. When the revolution angle is small in the L mode, the delays are shrunk as well as the corresponding chip thicknesses. On the contrary, the delays and chip thicknesses are enlarged for the R mode as the revolution angle is close to π . Both the earlier effect in L and the later effect in R can be almost enhanced to 10% when the cutting edge approaches the cutter tip. As a result, the average D2S, defined as a ratio of dynamic chip thickness to the corresponding static one, appears more different between using the proposed model and the HPA model in light-cut finishing. It is distinguished from the consistent performance in slot milling as shown in Fig. 9(b). The semi-discretization method has proved capable of determining system stabilities for a nonautonomous system in either a
Process parameters
Cutting force coefficients from orthogonal database
R = 4 mm
V m/min
l0 = 30 deg L = T = 15 deg f = 0.05 mm cf = 0.1 mm
Ct mm r = 0.40 + 0.0005V + 0.6Ct β = 26.8 − 0.0313V + 11.77Ct deg Tshear = 450.3 + 0.4V + 227.5Ct MPa
constant delay [19] or a periodic delay [14,20]. With this method, stabilities of light-cut finishing have been assessed using the existing model HPA and the proposed model. Dynamic properties of tool force coefficients as well as process parameters adopted in literature [17] were used here to examine the proposed chip thickness model, as listed in Table 3. The proposed chip thickness model can be validated by comparing the predicted stability lobes diagram (SLD) with that of experiments in literature [17], as depicted in Fig. 9(c). As the average time-varying delay is only 0.34% sooner than the pitch of the tooth, the differences against that of model HPA can be ignored. Another set of parameters, including f = 0.2 mm, cf = −0.2 mm, L = 15°, T = 0°, was carried out to show the differences between the proposed model and HPA, since a larger later delay can be captured. Although the average later effect of the delays is only about 2%, the right-shift effects of SLD have been observed for the proposed model as depicted in Fig. 9(d), which can draw a bigger different stability region for machining process optimization. This is because the locations of peak of SLD are defined by the ratio between the dominant natural frequency of the machine-tool system and the tooth passing frequency; and either the earlier effect or the later effect of time-varying delays
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can be amplified as the ratio changes. Therefore, the differences in dynamic performance between using the proposed model and the HPA model seem to be much more significant in comparison with that in static chip thickness modeling. 4. Conclusions In this paper, an in-depth geometric model is established to determine instantaneous chip thickness for a 5-axis ball-end finish milling. Applying the 3D3T expression of cutting edge, a numerical method which is able to handle cases with various feedrate, tool–workpiece inclination and cutter runout, is proposed. Then chip thickness around engagement boundaries is modeled directly to envisage the amplified overestimation effect in the finish process. In combination with the direct measuring chip volume of a commercial software package (Unigraphics), it is shown that one order reduction can be achieved for both the underestimated error in classical model SPA and the overestimated error in the widely used method ERF. Dynamic properties including timevarying delays, dynamic chip thicknesses and SLD are generated to reveal that the proposed model is also of a different nature compared with the existing model HPA. As the linear feed assumption is used for most conditions, the cutter inclinations for the adjacent two flutes are considered the same in the proposed chip thickness model. The relationship between previous cutting surface and current cutting tooth need to be modified for the case of 5-axis flow cut finishing or hole milling where component features are of the same scale with cutter dimensions, and time-varying delays might become more distinguished. In future work, modifications of local gesture transformation will be considered to describe the milling stabilities in 5-axis flow cut finishing. Acknowledgment This work was supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 50821003).
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