- Email: [email protected]
The effects of spindle vibration on surface generation in ultra-precision raster milling
International Journal of Machine Tools & Manufacture 71 (2013) 52–56
Contents lists available at SciVerse ScienceDirect
International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool
Short Communication
The effects of spindle vibration on surface generation in ultra-precision raster milling S.J. Zhang, S. To n State Key Laboratory in Ultra-precision Machining Technology, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong SAR, China
art ic l e i nf o
a b s t r a c t
Article history: Received 7 April 2013 Received in revised form 10 April 2013 Accepted 23 April 2013 Available online 29 April 2013
Spindle vibration has a significant influence on surface quality of ultra-precision-machined components. However, relatively few studies on the particular spindle vibration under the excitation of intermittent cutting forces in ultra-precision raster milling (UPRM) have been reported. In this study, a specialized model for an aerostatic bearing spindle under the impulsive excitation from intermittent cutting forces of UPRM is developed and its derived mathematical solutions reveal that the spindle vibration is impulsive response. The theoretical and experimental results signify that the impulsive spindle vibration produces inhomogeneous scallops forming ribbon-stripe patterns and irregular patterns like run-out on a surface of UPRM. The potential benefits for UPRM are the theoretical supports for optimization and prediction of surface generation through the optimal selection of spindle speed. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Modeling Spindle vibration Surface generation Ultra-precision raster milling
1. Introduction Ultra-precision raster milling (UPRM) is an advanced micromilling process for manufacturing freeform components. Since the micro-milling process is complex and the selections of cutting operations are multifarious [1], a lot of interest has been taken in studying the factors influencing surface generation. It is summarized that: cutting conditions (tool tip geometry, spindle speed, depth of cut, feed rate, swing distance, and step distance) [2–5], cutting strategies (horizontal cutting and vertical cutting) [4,5], shift length ratios [6], swelling and recovery [6], and tool wear [4] make a major impact on surface roughness; and cutting strategies [4], tool path generation [4,6], and kinematic errors of sliders [4] mainly influence the form accuracy of freeform surfaces. In most of the prior studies of surface generation in UPRM, the effects of cutting mechanism and material factors on surface generation have been overemphasized. Also, some researchers have studied the effects of spindle vibration on surface quality in micro-cutting. Martin et al. [7] discussed that the spindle vibration made an impact on surface topographies. Marsh [8] elaborated the effects of spindle dynamics on topographies of flat surfaces in precision fly-cutting. An et al. [9] experimentally and theoretically analyzed the tilting motions of the aerostatic bearing spindle influencing topographies of machined surfaces in ultra-precision fly cutting. However, the effects of the complex spindle dynamics on surface generation are not understood fully
n
Corresponding author. Tel.: +852 2766 6587; fax: +852 2764 7657. E-mail address: [email protected] (S. To).
0890-6955/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmachtools.2013.04.005
and completely. In our prior work [10,11], a complex five-degree-offreedom dynamic model for aerostatic bearing spindle vibration in ultra-precision diamond turning was developed to analyze its complex dynamics in order to study its effects on surface topography. Differently, the spindle in ultra-precision diamond turning is excited by the stable cutting forces, but the spindle in UPRM is acted on by intermittent cutting forces in UPRM. Nevertheless, the spindlevibration mechanism has not been understood fully in mathematics, and no work has been found on providing mathematical solutions for the particular spindle vibration under the excitation of intermittent cutting forces in UPRM. Especially, the effects of the particular spindle vibration on surface generation in UPRM have never been discussed. In this study, based on the previously-proposed five-degree-offreedom dynamic model for an aerostatic bearing spindle [11], the specialized model under the excitation of intermittent cutting forces of UPRM is established. More importantly, the mathematical solutions for the model are derived to analyze the effects of the particular spindle vibration on surface generation in UPRM, using surface generation technique. 2. Experimental setup A flat-cutting test of copper alloy was performed on an UPRM machine (Precitech Freeform 705G) (as shown in Fig. 1(a)) with an aerostatic bearing spindle (as shown in Fig. 1(b)) under the cutting conditions of Table 1 to generate a flat surface. In UPRM, a single point diamond tool is widely employed. As shown in Fig. 1(b), the
S.J. Zhang, S. To / International Journal of Machine Tools & Manufacture 71 (2013) 52–56
c1
C axis Spindle
k1
Shaft
X
Equilibrium distances
Y
B Z
k2 Tool
l1 c2 O,o
l2 Tilting distance
53
X/x,φ
e
θ ,Z/z m Mass of the shaft
d2 Raster cutting force k3
Y/ y,Ω
c3
Fr
Tool Thrust cutting force Ft
Fm Spin axis
Main cutting force Swing distance d1
Fig. 1. (a) The UPRM machine (Precitech Freeform 705G) with (b) an aerostatic bearing spindle and (c) its idealized model.
tool is mounted and rotated with the spindle shaft, and the workpiece is fixed with the B-table. Thus, the tool removes surface material at once per revolution, named intermittent cutting, and the efficient contacting time is also extremely short. More details are seen in Refs. [4,6]. In the milling process, since the spindle is acted on by intermittent cutting forces of UPRM during the extremely short contacting time, the intermittent cutting forces applied into the spindle are idealized as the impulsive excitation, under which the impulsive responses take place for the particular spindle vibration. The cutting forces were measured by a Kistler 9252A force sensor through three Kistler 5011B amplifiers. The Optical Profiling System (WYKO NT8000) was employed to measure the milled surface. 3. Modeling of spindle vibration 3.1. Dynamic model for spindle vibration In Fig. 1(c), the spindle shaft is seen as a rigid body whose mass is m. The spindle shaft is referred to and fixed with a reference system o(xyz), moving in an inertial coordinate system O(XYZ) (Equilibrium center). Its motions can be described by the rotations of o(xyz) around the z, x and y axes at the angles (θ, Ф, and
Table 1 Cutting conditions of raster milling. Spindle speed (ω) (rpm) Feed rate (fr) (mm/rev.) Depth of cut (do) (mm) Swing distance (d1) (mm) Step distance (sr) (mm) Tool nose radius (Rr) (mm) Tool rake angle (1) Front clearance angle (1) Cutting strategy Cutting mode
4900 20 5 28.48 10 0.619 0 15 Horizontal cutting Up-cutting
Ω which is equal to ωt, in which ω is spindle speed and t is time) in O(XYZ), and by the translations of o(xyz) along the X, Y and Z-axes at the displacements (x, y and z, respectively) in O(XYZ). _ ϕ _ and Ω, _ and the The corresponding angular velocities are θ, € ϕ€ and Ω, € respectively. corresponding angular accelerations are θ, The corresponding velocities are x_ , y_ and z_ , and the corresponding accelerations are x€ , y€ and z€ , respectively. In the spindle system, after balancing, the mass center coordi! nate in o(xyz) is e ¼ 0. The inertial tensor in o(xyz), and the
54
S.J. Zhang, S. To / International Journal of Machine Tools & Manufacture 71 (2013) 52–56
contacting time, the cutting forces and the corresponding torques in O(XYZ) are expressed in Eq. (1). 2
Jx 60 4 0
0 Jy 0
2 3 " # −F m F r d1 −F t d2 60 arccosðd1 −d0 =d1 Þ 6 7 7 0 5; τ ¼ ; 4 −F r 5 and F m d2 πω Jz −F t 0
3
In Eq. (3), the equations are coupled with each other. Hence, the natural frequencies of the tilting motions of the spindle shaft are written as below:
ð1Þ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u ðω21 þ ω22 þ ððJ y −J z Þ=J x ÞωððJ x −J z Þ=J y ÞωÞ2 −ðω1 ω2 Þ2 1 1 tω21 þ ω22 þ ððJ y −J z Þ=J x ÞωððJ x −J z Þ=J y Þω ωϕ ¼ − fϕ ¼ 2π 2π 2 2
ð5Þ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u ðω21 þ ω22 þ ððJ y −J z Þ=J x ÞωððJ x −J z Þ=J y ÞωÞ2 −ðω1 ω2 Þ2 1 1 tω21 þ ω22 þ ððJ y −J z Þ=J x ÞωððJ x −J z Þ=J y Þω ωθ ¼ þ fθ ¼ 2π 2π 2 2
where the contacting time of intermittent cutting forces is further less than the vibration period. Since these perturbation angles are extremely tiny due to micrometric bearing clearance, sin ϕ≈ϕ, sin θ≈θ, cos ϕ≈1 and cos θ≈1( named linear small quantity) and _ _ _ _ θθ≈0, θϕ≈0, ϕϕ≈0, ϕθ≈0, ϕ sin ϕ≈0, θ sin θ≈0, ϕ sin θ≈0 and θ sin ϕ≈0 (named quadratic small quantity). According to Ref. [11], the specialized model for the particular spindle vibration of UPRM is expressed as: 8 € _ > < my þ c1 y þ k1 y ¼ −F r mx€ þ c2 x_ þ k2 x ¼ −F m ð2Þ > : mz€ þ c z_ þ k z ¼ −F t 3 3 8 € x −J y Þωϕ_ þ dz θ_ ¼ −ðk2 þ k3 Þðl1 2 þ l2 2 Þθ−k1 R2 θ þ F m d2 < J z θ−ðJ : J x ϕ€ þ ðJ z −J y Þωθ_ þ dx ϕ_ ¼ −ðk2 þ k3 Þðl1 þ l2 Þϕ−k1 R ϕ þ F r d1 −F t d2 2
2
2
ð3Þ 2
where, R is the radius of the spindle shaft, −ðk2 þ k3 Þðl1 þ 2 2 2 l2 Þθ−k1 R2 θ and −ðk2 þ k3 Þðl1 þ l2 Þϕ−k1 R2 ϕ are represented as the rotational torques around the x-axial direction and the z-axial direction, respectively. As the damping ratio of the pressurized air film is further tiny, the resistance and the drag torques can be negligible. Therefore, the spindle system is subjected to intermittent cutting forces and intermittent torques without inertial forces ! and inertial torques since e ¼ 0. Due to the extremely short contacting time, the responses of the spindle applied by such excitation are referred to as impulsive responses. 3.2. Mathematical solutions Since the contacting time is extremely short further less than the spindle vibration period, the vibration is idealized as the impulse spindle vibration. For Eq. (2), the general solutions can be obtained as a sequential set of Eq. (4) according to Ref. [11]. 8 _ r τ=mÞ yðtÞ ¼ yðiTÞ cos ðωz ðt−iTÞÞ þ yðiTÞþð−F sin ðωy ðt−iTÞÞ > > ωy > < x_ ðiTÞþð−F m τ=mÞ xðtÞ ¼ xðiTÞ cos ðωz ðt−iTÞÞ þ sin ðωx ðt−iTÞÞ ð4Þ ωx > pffiffiffiffiffiffiffiffiffiffiffi > > z_ ðiTÞþð−F t τ=mÞ 2 : sin ðωz 1−μ ðt−iTÞÞ zðtÞ ¼ zðiTÞ cos ðωz ðt−iTÞÞ þ ωz where, t∈ðiTði þ 1ÞT, i¼ 0, 1, , N, 8 > < c1 ¼ 0 c2 ¼ 0 , > :c ¼0 3
8 y_ ð0Þ ¼ −Fmr τ ; yð0Þ ¼ 0 > > < x_ ð0Þ ¼ −Fmm τ ; xð0Þ ¼ 0 , > > : z_ ð0Þ ¼ −F t τ ; zð0Þ ¼ 0 m
8 > > >fy ¼ > > < fx ¼ > > > > > : fz ¼
1 2π ωy
¼
1 2π
1 2π ωx
¼
1 2π
1 2π ωz
¼
1 2π
qffiffiffiffi k1 m
qffiffiffiffi k2 m
qffiffiffiffi k3 m
(the axial frequency (AF), the radial frequencies (RFs)), and T is one period ofs spindle rotation.
ð6Þ
The above frequencies are named coupled tilting frequencies (CTFs) being influenced by the spindle speed. Since the dynamic responses of the system are the impulsive responses, the solutions of Eq. (3) only contain the homogeneous solutions without the particular solutions. Thus, the homogeneous solutions are in the form of sine and cosine functions at the CTFs. Therefore, the general solutions are set as Eq. (7). Also, substituting Eq. (7) with the corresponding first and second order of differential equations for Eq. (7) into (3), respectively, the general solutions with the corresponding parameters are obtained as Eq. (7). (
ϕðtÞ ¼ A1 sin ωθ ðt−iTÞ þ A2 cos ωθ ðt−iTÞ þ A3 sin ωϕ ðt−iTÞ þ A4 cos ωϕ ðt−iTÞ θðtÞ ¼ B1 cos ωθ ðt−iTÞ þ B2 sin ωθ ðt−iTÞ þ B3 cos ωϕ ðt−iTÞ þ B4 sin ωϕ ðt−iTÞ
ð7Þ where t∈ðiTði þ 1ÞT; Cz ¼
ðJ x −J y Þω ; Jz
Dx ¼
dx dz ¼ Dz ¼ ¼ 0; Jx Jz
θð0Þ ¼ 0;
F r d1 −F t d2 _ ; ϕð0Þ ¼ Jx 2
ϕð0Þ ¼ 0;
ðJ z −J y Þω ; Jx
F m d2 _ θð0Þ ¼ ; Jz 2
ω21 ¼
Cx ¼
2
ððk2 þ k3 Þðl1 þ l2 Þ þ k1 R2 Þ ; Jz
2
ððk2 þ k3 Þðl1 þ l2 Þ þ k1 R2 Þ F r d1 −F t d2 _ ¼ iTÞ ¼ ϕðiTÞ _ ; ϕðt þ ; Jx Jx F r d1 −F t d2 _ þ A1 ¼ −θðiTÞðω2θ −ω21 Þðω2ϕ −ω21 Þ þ C z ðω21 −ω2θ Þ ϕðiTÞ Jx F m d2 _ ¼ iTÞ ¼ θðiTÞ _ þ C z ωθ ðω2ϕ −ω2θ Þ ; θðt Jz F d ω m 2 θ 2 2 _ A2 ¼ ϕðiTÞ− θðiTÞðω ðω2θ −ω21 Þ þ C z ω2θ ϕðiTÞ ðω2ϕ −ω21 Þ ϕ −ω1 Þ þ Jz . C z ðω2θ −ω2ϕ Þω21 F r d1 −F t d2 _ þ A3 ¼ θðiTÞðω2θ −ω21 Þðω2ϕ −ω21 Þ þ C z ðω2ϕ −ω21 Þ ϕðiTÞ Jx = C z ωϕ ðω2ϕ −ω2θ Þ
ω22 ¼
F m d2 ωθ 2 2 2 2 _ A4 ¼ θðiTÞðω ðωθ −ω1 Þ þ C z ω2θ ϕðiTÞ ðω2ϕ −ω21 Þ= C z ðω2θ −ω2ϕ Þω21 ϕ −ω1 Þ þ Jz
F r d1 −F t d2 _ B1 ¼ θðiTÞðω2ϕ −ω21 Þ þ C z ϕðiTÞ =ðω2ϕ −ω2θ Þ þ Jx
F m d2 ωθ 2 2 2 2 2 2 2 _ B2 ¼ θðiTÞðω ωθ ðω1 −ωϕ Þ−C z ω2ϕ ω2θ ϕðiTÞ = ωθ ðω2θ −ω2ϕ Þω21 θ ω1 −ωϕ ωϕ Þ þ Jz
F r d1 −F t d2 _ B3 ¼ − θðiTÞðω2θ −ω21 Þ þ C z ðϕðiTÞ þ Þ =ðω2ϕ −ω2θ Þ; and Jx F d ω m 2 θ 2 2 _ B4 ¼ θðiTÞðω ðω2θ −ω21 Þ þ C z ω2θ ϕðiTÞ ωϕ = ðω2θ −ω2ϕ Þω21 ϕ −ω1 Þ þ Jz
S.J. Zhang, S. To / International Journal of Machine Tools & Manufacture 71 (2013) 52–56
55
Eqs. (4) and (7) mathematically represent the dynamic responses of the spindle excited by intermittent cutting forces of UPRM at spindle speed ω. The results are obtained as follows. (i) The responses are periodical impulsive responses. (ii) The impulsive responses of the spindle under the excitation of intermittent cutting forces portrait the relative distances between a tool and a workpiece, and the relative distances in each contacting time together form spindle-vibration-induced profiles (SVIPs) along the feed direction, which are irregular and transilient. (iii) These SVIPs generate inhomogeneous scallops which can form irregular patterns and ribbon-stripe patterns through tool loci. (iv) The equations also express the frequency characteristics of the aerostatic bearing spindle that possesses the AF, RFs and CTFs. (v) The mathematical solutions also provide a potential approach and theoretical basis to an investigation into the effects of the particular spindle vibration on surface generation for practical applications in UPRM such as surface prediction and machining optimization through the optimal selection of spindle speed, since a synchronous impulsive excitation will synchronous vibration of spindle, whose frequency is determined by spindle speed. In this study, a surface generation model developed by Cheung et al. [2] is employed to analyze the effects of the particular spindle vibration on surface generation of UPRM. The impulsive responses of the particular spindle vibration are input into the surface generation model proposed by Cheung to generate surface topography. The simulated surface topography generated under the cutting conditions of Table 1 is shown in Fig. 2(a). In addition, a flat surface milled under the same cutting conditions was carried out on the employed UPRM machine to identify the effects of the particular spindle vibration on surface generation. The milled surface was measured by the Optical Profiling System NT8000, as shown in Fig. 2(b–d). Both of the simulated and measured surface topographies clearly show inhomogeneous scallops formed thought tool loci on the milled surface. In Fig. 2(a) and (b), some ribbon stripes along the feed direction indicated by the square-dot arrows are observed in the raster direction. These ribbon stripes were generated by the repetition of the SVIPs along the feed direction. Also, it is obviously observed that the SVIPs change sharply, as indicated by the round-dot arrows, as shown in Fig. 2(c) that plots one profile along the feed direction and in Fig. 2(d) that depicts one profile along the raster direction. Hence, the SVIPs produced the patterns through the inhomogeneous scallops or tool loci generated at the milled surface. Additionally, the more irregular patterns or irregular tool loci referred to as the run-out [5] are also detected, as obviously shown in Fig. 2(a) and (b), because the SVIPs were irregular and transilient and determined by the impulsive responses for the particular spindle vibration influenced by spindle speed. Therefore, the irregular patterns of the surface topography were produced by the particular spindle vibration. It also explains why the run-out takes place on the milled surface. Furthermore, there are some factors influencing surface topography. Fig. 2(c) and (d) obviously shows the material pile-up at the height of 2 nm along the feed direction and at the height of 17 nm along the raster direction. According to Ref. [6], the material swelling and recovery is at the height of 20 nm. In Fig. 2(b), the residual height of the measured surface is 35.9 nm without considering the heights of the material pile-up and the material swelling and recovery. Additionally, the ideal surface height according to Ref. [5] is 20.2 nm, which is closely relevant to the height of the material pile-up. The residual height is close to the height of the SVIPs at 34 nm calculated by Eqs. (4) and (7), which is the same to the height of the simulated surface obtained from
Raster direction Y/mm
4. Results and discussion
Feed direction X/mm
Material pile-up
Feed direction X/mm
Material pile-up
Raster direction Y/μm Fig. 2. Surface generation under the cutting conditions: (a) simulated topography and (b) measured topography with (c) one profile along the feed direction and (d) one profile along the raster direction.
Fig. 2(a) without the ideal surface height. Hence, the contribution of the spindle vibration under the cutting conditions is 45%. It also
56
S.J. Zhang, S. To / International Journal of Machine Tools & Manufacture 71 (2013) 52–56
explores that the particular spindle vibration in UPRM plays a crucial role in the factors on surface generation. And the mathematical solutions can be employed to predict the effects of the particular spindle vibration of UPRM on surface quality. Overall, the experimental observation with the simulated surface topography well supports the effects of the particular spindle vibration of UPRM on surface generation based on the derived mathematical solutions for the specialized dynamic model. 5. Conclusions In this paper, a specialized dynamic model for the particular spindle vibration under the intermittent cutting forces of ultraprecision raster milling (UPRM) is developed. Its corresponding mathematical solutions are derived to study the effects of the particular spindle vibration in UPRM on surface generation. The effects of the particular spindle vibration on surface generation have been confirmed by a milled surface with a simulated surface topography. The results are found as follows: (i) The spindle excited under the intermittent cutting forces of UPRM at the spindle speed vibrates at the axial frequency (AF), radial frequencies (RFs) and coupled tilting frequencies (CTFs). The particular spindle vibration is the intermittent impulsive responses. (ii) The solutions theoretically support that the particular spindle vibration under the intermittent cutting forces of UPRM produces the inhomogeneous scallops forming the ribbonstripe and irregular patterns, which are made out by the simulated and measured surface topographies generated in UPRM, and explain that the run-out phenomenon taking place on the milled surface through the irregular tool loci is induced by the spindle vibration under the intermittent cutting forces of UPRM. (iii) Since the particular spindle vibration is the periodical impulsive responses, it is interesting to note that the excitation frequency of the intermittent cutting forces, i.e. spindle speed, would influence the intermittent impulsive responses. (iv) Further, it is a potential means to improving the surface quality through the optimal selection of spindle speed. This will contribute significantly to the further improvement for
surface quality of ultra-precision raster milling, considering the effects of spindle vibration.
Acknowledgment This work was supported by The Research Grants Council of the Hong Kong Special Administrative Region of the People's Republic of China (Project No. PolyU 5287/10E). References [1] C.F. Cheung, W.B. Lee, S. To, A framework of a model-based simulation system for predicting surface generation in ultra-precision raster milling of freeform surfaces, in: Proceedings of American Society for Precision Engineering Annual Meeting, Orlando, Florida, October 24–29, 2004. [2] C.F. Cheung, L.B. Kong, W.B. Lee, S. To, Modelling and simulation of freeform surface generation in ultra-precision raster milling, Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture 220 (2006) 1787–1801. [3] M.N. Cheng, C.F. Cheung, W.B. Lee, S. To, L.B. Kong, Theoretical and experimental analysis of nano-surface generation in ultra-precision raster milling, International Journal of Machine Tools and Manufacture 48 (2008) 1090–1102. [4] L.B. Kong, Modeling of Ultra-precision Raster Milling and Characterization of Optical Freeform Surface. Ph.D. Dissertation, The Hong Kong Polytechnic University, Hong Kong, 2009. [5] M.N. Cheng, Optimization of Surface Generation in Ultra-Precision Multi-axis Raster Milling. M.Phil. Thesis, The Hong Kong Polytechnic University, Hong Kong, 2006. [6] Sujuan Wang, Modelling and Optimization of Cutting Strategy and Surface Generation in Ultra-Precision Raster Milling. Ph.D. Dissertation, The Hong Kong Polytechnic University, Hong Kong, 2010. [7] D.L. Martin, A.N. Tabenkin, F.G. Parsons, Precision spindle and bearing error analysis, International Journal of Machine Tools and Manufacture 35 (1995) 187–193. [8] E.R. Marsh, D.A. Arneson, M.J. Van Doren, S.A. Blystone, The effects of spindle dynamics on precision flycutting, in: Proceedings of American Society for Precision Engineering Annual Meeting, Norfolk, Virgina, October 09–14, 2005. [9] C.H. An, Y. Zhang, Q. Xu, F.H. Zhang, J.F. Zhang, L.J. Zhang, J.H. Wang, Modeling of dynamic characteristic of the aerostatic bearing spindle in an ultraprecision fly cutting machine, International Journal of Machine Tools and Manufacture 50 (2010) 374–385. [10] S.J. Zhang, S. To, C.F. Cheung, H.T. Wang, Dynamic characteristics of an aerostatic bearing spindle and its influence on surface topography in ultraprecision diamond turning, International Journal of Machine Tools and Manufacture 62 (2012) 1–12. [11] S.J. Zhang, S. To, H.T. Wang, A theoretical and experimental investigation into five-DOF dynamic characteristics of an aerostatic bearing spindle in ultraprecision diamond turning, International Journal of Machine Tools and Manufacture 71 (2013) 1–10.
Contents lists available at SciVerse ScienceDirect
International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool
Short Communication
The effects of spindle vibration on surface generation in ultra-precision raster milling S.J. Zhang, S. To n State Key Laboratory in Ultra-precision Machining Technology, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong SAR, China
art ic l e i nf o
a b s t r a c t
Article history: Received 7 April 2013 Received in revised form 10 April 2013 Accepted 23 April 2013 Available online 29 April 2013
Spindle vibration has a significant influence on surface quality of ultra-precision-machined components. However, relatively few studies on the particular spindle vibration under the excitation of intermittent cutting forces in ultra-precision raster milling (UPRM) have been reported. In this study, a specialized model for an aerostatic bearing spindle under the impulsive excitation from intermittent cutting forces of UPRM is developed and its derived mathematical solutions reveal that the spindle vibration is impulsive response. The theoretical and experimental results signify that the impulsive spindle vibration produces inhomogeneous scallops forming ribbon-stripe patterns and irregular patterns like run-out on a surface of UPRM. The potential benefits for UPRM are the theoretical supports for optimization and prediction of surface generation through the optimal selection of spindle speed. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Modeling Spindle vibration Surface generation Ultra-precision raster milling
1. Introduction Ultra-precision raster milling (UPRM) is an advanced micromilling process for manufacturing freeform components. Since the micro-milling process is complex and the selections of cutting operations are multifarious [1], a lot of interest has been taken in studying the factors influencing surface generation. It is summarized that: cutting conditions (tool tip geometry, spindle speed, depth of cut, feed rate, swing distance, and step distance) [2–5], cutting strategies (horizontal cutting and vertical cutting) [4,5], shift length ratios [6], swelling and recovery [6], and tool wear [4] make a major impact on surface roughness; and cutting strategies [4], tool path generation [4,6], and kinematic errors of sliders [4] mainly influence the form accuracy of freeform surfaces. In most of the prior studies of surface generation in UPRM, the effects of cutting mechanism and material factors on surface generation have been overemphasized. Also, some researchers have studied the effects of spindle vibration on surface quality in micro-cutting. Martin et al. [7] discussed that the spindle vibration made an impact on surface topographies. Marsh [8] elaborated the effects of spindle dynamics on topographies of flat surfaces in precision fly-cutting. An et al. [9] experimentally and theoretically analyzed the tilting motions of the aerostatic bearing spindle influencing topographies of machined surfaces in ultra-precision fly cutting. However, the effects of the complex spindle dynamics on surface generation are not understood fully
n
Corresponding author. Tel.: +852 2766 6587; fax: +852 2764 7657. E-mail address: [email protected] (S. To).
0890-6955/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmachtools.2013.04.005
and completely. In our prior work [10,11], a complex five-degree-offreedom dynamic model for aerostatic bearing spindle vibration in ultra-precision diamond turning was developed to analyze its complex dynamics in order to study its effects on surface topography. Differently, the spindle in ultra-precision diamond turning is excited by the stable cutting forces, but the spindle in UPRM is acted on by intermittent cutting forces in UPRM. Nevertheless, the spindlevibration mechanism has not been understood fully in mathematics, and no work has been found on providing mathematical solutions for the particular spindle vibration under the excitation of intermittent cutting forces in UPRM. Especially, the effects of the particular spindle vibration on surface generation in UPRM have never been discussed. In this study, based on the previously-proposed five-degree-offreedom dynamic model for an aerostatic bearing spindle [11], the specialized model under the excitation of intermittent cutting forces of UPRM is established. More importantly, the mathematical solutions for the model are derived to analyze the effects of the particular spindle vibration on surface generation in UPRM, using surface generation technique. 2. Experimental setup A flat-cutting test of copper alloy was performed on an UPRM machine (Precitech Freeform 705G) (as shown in Fig. 1(a)) with an aerostatic bearing spindle (as shown in Fig. 1(b)) under the cutting conditions of Table 1 to generate a flat surface. In UPRM, a single point diamond tool is widely employed. As shown in Fig. 1(b), the
S.J. Zhang, S. To / International Journal of Machine Tools & Manufacture 71 (2013) 52–56
c1
C axis Spindle
k1
Shaft
X
Equilibrium distances
Y
B Z
k2 Tool
l1 c2 O,o
l2 Tilting distance
53
X/x,φ
e
θ ,Z/z m Mass of the shaft
d2 Raster cutting force k3
Y/ y,Ω
c3
Fr
Tool Thrust cutting force Ft
Fm Spin axis
Main cutting force Swing distance d1
Fig. 1. (a) The UPRM machine (Precitech Freeform 705G) with (b) an aerostatic bearing spindle and (c) its idealized model.
tool is mounted and rotated with the spindle shaft, and the workpiece is fixed with the B-table. Thus, the tool removes surface material at once per revolution, named intermittent cutting, and the efficient contacting time is also extremely short. More details are seen in Refs. [4,6]. In the milling process, since the spindle is acted on by intermittent cutting forces of UPRM during the extremely short contacting time, the intermittent cutting forces applied into the spindle are idealized as the impulsive excitation, under which the impulsive responses take place for the particular spindle vibration. The cutting forces were measured by a Kistler 9252A force sensor through three Kistler 5011B amplifiers. The Optical Profiling System (WYKO NT8000) was employed to measure the milled surface. 3. Modeling of spindle vibration 3.1. Dynamic model for spindle vibration In Fig. 1(c), the spindle shaft is seen as a rigid body whose mass is m. The spindle shaft is referred to and fixed with a reference system o(xyz), moving in an inertial coordinate system O(XYZ) (Equilibrium center). Its motions can be described by the rotations of o(xyz) around the z, x and y axes at the angles (θ, Ф, and
Table 1 Cutting conditions of raster milling. Spindle speed (ω) (rpm) Feed rate (fr) (mm/rev.) Depth of cut (do) (mm) Swing distance (d1) (mm) Step distance (sr) (mm) Tool nose radius (Rr) (mm) Tool rake angle (1) Front clearance angle (1) Cutting strategy Cutting mode
4900 20 5 28.48 10 0.619 0 15 Horizontal cutting Up-cutting
Ω which is equal to ωt, in which ω is spindle speed and t is time) in O(XYZ), and by the translations of o(xyz) along the X, Y and Z-axes at the displacements (x, y and z, respectively) in O(XYZ). _ ϕ _ and Ω, _ and the The corresponding angular velocities are θ, € ϕ€ and Ω, € respectively. corresponding angular accelerations are θ, The corresponding velocities are x_ , y_ and z_ , and the corresponding accelerations are x€ , y€ and z€ , respectively. In the spindle system, after balancing, the mass center coordi! nate in o(xyz) is e ¼ 0. The inertial tensor in o(xyz), and the
54
S.J. Zhang, S. To / International Journal of Machine Tools & Manufacture 71 (2013) 52–56
contacting time, the cutting forces and the corresponding torques in O(XYZ) are expressed in Eq. (1). 2
Jx 60 4 0
0 Jy 0
2 3 " # −F m F r d1 −F t d2 60 arccosðd1 −d0 =d1 Þ 6 7 7 0 5; τ ¼ ; 4 −F r 5 and F m d2 πω Jz −F t 0
3
In Eq. (3), the equations are coupled with each other. Hence, the natural frequencies of the tilting motions of the spindle shaft are written as below:
ð1Þ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u ðω21 þ ω22 þ ððJ y −J z Þ=J x ÞωððJ x −J z Þ=J y ÞωÞ2 −ðω1 ω2 Þ2 1 1 tω21 þ ω22 þ ððJ y −J z Þ=J x ÞωððJ x −J z Þ=J y Þω ωϕ ¼ − fϕ ¼ 2π 2π 2 2
ð5Þ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u ðω21 þ ω22 þ ððJ y −J z Þ=J x ÞωððJ x −J z Þ=J y ÞωÞ2 −ðω1 ω2 Þ2 1 1 tω21 þ ω22 þ ððJ y −J z Þ=J x ÞωððJ x −J z Þ=J y Þω ωθ ¼ þ fθ ¼ 2π 2π 2 2
where the contacting time of intermittent cutting forces is further less than the vibration period. Since these perturbation angles are extremely tiny due to micrometric bearing clearance, sin ϕ≈ϕ, sin θ≈θ, cos ϕ≈1 and cos θ≈1( named linear small quantity) and _ _ _ _ θθ≈0, θϕ≈0, ϕϕ≈0, ϕθ≈0, ϕ sin ϕ≈0, θ sin θ≈0, ϕ sin θ≈0 and θ sin ϕ≈0 (named quadratic small quantity). According to Ref. [11], the specialized model for the particular spindle vibration of UPRM is expressed as: 8 € _ > < my þ c1 y þ k1 y ¼ −F r mx€ þ c2 x_ þ k2 x ¼ −F m ð2Þ > : mz€ þ c z_ þ k z ¼ −F t 3 3 8 € x −J y Þωϕ_ þ dz θ_ ¼ −ðk2 þ k3 Þðl1 2 þ l2 2 Þθ−k1 R2 θ þ F m d2 < J z θ−ðJ : J x ϕ€ þ ðJ z −J y Þωθ_ þ dx ϕ_ ¼ −ðk2 þ k3 Þðl1 þ l2 Þϕ−k1 R ϕ þ F r d1 −F t d2 2
2
2
ð3Þ 2
where, R is the radius of the spindle shaft, −ðk2 þ k3 Þðl1 þ 2 2 2 l2 Þθ−k1 R2 θ and −ðk2 þ k3 Þðl1 þ l2 Þϕ−k1 R2 ϕ are represented as the rotational torques around the x-axial direction and the z-axial direction, respectively. As the damping ratio of the pressurized air film is further tiny, the resistance and the drag torques can be negligible. Therefore, the spindle system is subjected to intermittent cutting forces and intermittent torques without inertial forces ! and inertial torques since e ¼ 0. Due to the extremely short contacting time, the responses of the spindle applied by such excitation are referred to as impulsive responses. 3.2. Mathematical solutions Since the contacting time is extremely short further less than the spindle vibration period, the vibration is idealized as the impulse spindle vibration. For Eq. (2), the general solutions can be obtained as a sequential set of Eq. (4) according to Ref. [11]. 8 _ r τ=mÞ yðtÞ ¼ yðiTÞ cos ðωz ðt−iTÞÞ þ yðiTÞþð−F sin ðωy ðt−iTÞÞ > > ωy > < x_ ðiTÞþð−F m τ=mÞ xðtÞ ¼ xðiTÞ cos ðωz ðt−iTÞÞ þ sin ðωx ðt−iTÞÞ ð4Þ ωx > pffiffiffiffiffiffiffiffiffiffiffi > > z_ ðiTÞþð−F t τ=mÞ 2 : sin ðωz 1−μ ðt−iTÞÞ zðtÞ ¼ zðiTÞ cos ðωz ðt−iTÞÞ þ ωz where, t∈ðiTði þ 1ÞT, i¼ 0, 1, , N, 8 > < c1 ¼ 0 c2 ¼ 0 , > :c ¼0 3
8 y_ ð0Þ ¼ −Fmr τ ; yð0Þ ¼ 0 > > < x_ ð0Þ ¼ −Fmm τ ; xð0Þ ¼ 0 , > > : z_ ð0Þ ¼ −F t τ ; zð0Þ ¼ 0 m
8 > > >fy ¼ > > < fx ¼ > > > > > : fz ¼
1 2π ωy
¼
1 2π
1 2π ωx
¼
1 2π
1 2π ωz
¼
1 2π
qffiffiffiffi k1 m
qffiffiffiffi k2 m
qffiffiffiffi k3 m
(the axial frequency (AF), the radial frequencies (RFs)), and T is one period ofs spindle rotation.
ð6Þ
The above frequencies are named coupled tilting frequencies (CTFs) being influenced by the spindle speed. Since the dynamic responses of the system are the impulsive responses, the solutions of Eq. (3) only contain the homogeneous solutions without the particular solutions. Thus, the homogeneous solutions are in the form of sine and cosine functions at the CTFs. Therefore, the general solutions are set as Eq. (7). Also, substituting Eq. (7) with the corresponding first and second order of differential equations for Eq. (7) into (3), respectively, the general solutions with the corresponding parameters are obtained as Eq. (7). (
ϕðtÞ ¼ A1 sin ωθ ðt−iTÞ þ A2 cos ωθ ðt−iTÞ þ A3 sin ωϕ ðt−iTÞ þ A4 cos ωϕ ðt−iTÞ θðtÞ ¼ B1 cos ωθ ðt−iTÞ þ B2 sin ωθ ðt−iTÞ þ B3 cos ωϕ ðt−iTÞ þ B4 sin ωϕ ðt−iTÞ
ð7Þ where t∈ðiTði þ 1ÞT; Cz ¼
ðJ x −J y Þω ; Jz
Dx ¼
dx dz ¼ Dz ¼ ¼ 0; Jx Jz
θð0Þ ¼ 0;
F r d1 −F t d2 _ ; ϕð0Þ ¼ Jx 2
ϕð0Þ ¼ 0;
ðJ z −J y Þω ; Jx
F m d2 _ θð0Þ ¼ ; Jz 2
ω21 ¼
Cx ¼
2
ððk2 þ k3 Þðl1 þ l2 Þ þ k1 R2 Þ ; Jz
2
ððk2 þ k3 Þðl1 þ l2 Þ þ k1 R2 Þ F r d1 −F t d2 _ ¼ iTÞ ¼ ϕðiTÞ _ ; ϕðt þ ; Jx Jx F r d1 −F t d2 _ þ A1 ¼ −θðiTÞðω2θ −ω21 Þðω2ϕ −ω21 Þ þ C z ðω21 −ω2θ Þ ϕðiTÞ Jx F m d2 _ ¼ iTÞ ¼ θðiTÞ _ þ C z ωθ ðω2ϕ −ω2θ Þ ; θðt Jz F d ω m 2 θ 2 2 _ A2 ¼ ϕðiTÞ− θðiTÞðω ðω2θ −ω21 Þ þ C z ω2θ ϕðiTÞ ðω2ϕ −ω21 Þ ϕ −ω1 Þ þ Jz . C z ðω2θ −ω2ϕ Þω21 F r d1 −F t d2 _ þ A3 ¼ θðiTÞðω2θ −ω21 Þðω2ϕ −ω21 Þ þ C z ðω2ϕ −ω21 Þ ϕðiTÞ Jx = C z ωϕ ðω2ϕ −ω2θ Þ
ω22 ¼
F m d2 ωθ 2 2 2 2 _ A4 ¼ θðiTÞðω ðωθ −ω1 Þ þ C z ω2θ ϕðiTÞ ðω2ϕ −ω21 Þ= C z ðω2θ −ω2ϕ Þω21 ϕ −ω1 Þ þ Jz
F r d1 −F t d2 _ B1 ¼ θðiTÞðω2ϕ −ω21 Þ þ C z ϕðiTÞ =ðω2ϕ −ω2θ Þ þ Jx
F m d2 ωθ 2 2 2 2 2 2 2 _ B2 ¼ θðiTÞðω ωθ ðω1 −ωϕ Þ−C z ω2ϕ ω2θ ϕðiTÞ = ωθ ðω2θ −ω2ϕ Þω21 θ ω1 −ωϕ ωϕ Þ þ Jz
F r d1 −F t d2 _ B3 ¼ − θðiTÞðω2θ −ω21 Þ þ C z ðϕðiTÞ þ Þ =ðω2ϕ −ω2θ Þ; and Jx F d ω m 2 θ 2 2 _ B4 ¼ θðiTÞðω ðω2θ −ω21 Þ þ C z ω2θ ϕðiTÞ ωϕ = ðω2θ −ω2ϕ Þω21 ϕ −ω1 Þ þ Jz
S.J. Zhang, S. To / International Journal of Machine Tools & Manufacture 71 (2013) 52–56
55
Eqs. (4) and (7) mathematically represent the dynamic responses of the spindle excited by intermittent cutting forces of UPRM at spindle speed ω. The results are obtained as follows. (i) The responses are periodical impulsive responses. (ii) The impulsive responses of the spindle under the excitation of intermittent cutting forces portrait the relative distances between a tool and a workpiece, and the relative distances in each contacting time together form spindle-vibration-induced profiles (SVIPs) along the feed direction, which are irregular and transilient. (iii) These SVIPs generate inhomogeneous scallops which can form irregular patterns and ribbon-stripe patterns through tool loci. (iv) The equations also express the frequency characteristics of the aerostatic bearing spindle that possesses the AF, RFs and CTFs. (v) The mathematical solutions also provide a potential approach and theoretical basis to an investigation into the effects of the particular spindle vibration on surface generation for practical applications in UPRM such as surface prediction and machining optimization through the optimal selection of spindle speed, since a synchronous impulsive excitation will synchronous vibration of spindle, whose frequency is determined by spindle speed. In this study, a surface generation model developed by Cheung et al. [2] is employed to analyze the effects of the particular spindle vibration on surface generation of UPRM. The impulsive responses of the particular spindle vibration are input into the surface generation model proposed by Cheung to generate surface topography. The simulated surface topography generated under the cutting conditions of Table 1 is shown in Fig. 2(a). In addition, a flat surface milled under the same cutting conditions was carried out on the employed UPRM machine to identify the effects of the particular spindle vibration on surface generation. The milled surface was measured by the Optical Profiling System NT8000, as shown in Fig. 2(b–d). Both of the simulated and measured surface topographies clearly show inhomogeneous scallops formed thought tool loci on the milled surface. In Fig. 2(a) and (b), some ribbon stripes along the feed direction indicated by the square-dot arrows are observed in the raster direction. These ribbon stripes were generated by the repetition of the SVIPs along the feed direction. Also, it is obviously observed that the SVIPs change sharply, as indicated by the round-dot arrows, as shown in Fig. 2(c) that plots one profile along the feed direction and in Fig. 2(d) that depicts one profile along the raster direction. Hence, the SVIPs produced the patterns through the inhomogeneous scallops or tool loci generated at the milled surface. Additionally, the more irregular patterns or irregular tool loci referred to as the run-out [5] are also detected, as obviously shown in Fig. 2(a) and (b), because the SVIPs were irregular and transilient and determined by the impulsive responses for the particular spindle vibration influenced by spindle speed. Therefore, the irregular patterns of the surface topography were produced by the particular spindle vibration. It also explains why the run-out takes place on the milled surface. Furthermore, there are some factors influencing surface topography. Fig. 2(c) and (d) obviously shows the material pile-up at the height of 2 nm along the feed direction and at the height of 17 nm along the raster direction. According to Ref. [6], the material swelling and recovery is at the height of 20 nm. In Fig. 2(b), the residual height of the measured surface is 35.9 nm without considering the heights of the material pile-up and the material swelling and recovery. Additionally, the ideal surface height according to Ref. [5] is 20.2 nm, which is closely relevant to the height of the material pile-up. The residual height is close to the height of the SVIPs at 34 nm calculated by Eqs. (4) and (7), which is the same to the height of the simulated surface obtained from
Raster direction Y/mm
4. Results and discussion
Feed direction X/mm
Material pile-up
Feed direction X/mm
Material pile-up
Raster direction Y/μm Fig. 2. Surface generation under the cutting conditions: (a) simulated topography and (b) measured topography with (c) one profile along the feed direction and (d) one profile along the raster direction.
Fig. 2(a) without the ideal surface height. Hence, the contribution of the spindle vibration under the cutting conditions is 45%. It also
56
S.J. Zhang, S. To / International Journal of Machine Tools & Manufacture 71 (2013) 52–56
explores that the particular spindle vibration in UPRM plays a crucial role in the factors on surface generation. And the mathematical solutions can be employed to predict the effects of the particular spindle vibration of UPRM on surface quality. Overall, the experimental observation with the simulated surface topography well supports the effects of the particular spindle vibration of UPRM on surface generation based on the derived mathematical solutions for the specialized dynamic model. 5. Conclusions In this paper, a specialized dynamic model for the particular spindle vibration under the intermittent cutting forces of ultraprecision raster milling (UPRM) is developed. Its corresponding mathematical solutions are derived to study the effects of the particular spindle vibration in UPRM on surface generation. The effects of the particular spindle vibration on surface generation have been confirmed by a milled surface with a simulated surface topography. The results are found as follows: (i) The spindle excited under the intermittent cutting forces of UPRM at the spindle speed vibrates at the axial frequency (AF), radial frequencies (RFs) and coupled tilting frequencies (CTFs). The particular spindle vibration is the intermittent impulsive responses. (ii) The solutions theoretically support that the particular spindle vibration under the intermittent cutting forces of UPRM produces the inhomogeneous scallops forming the ribbonstripe and irregular patterns, which are made out by the simulated and measured surface topographies generated in UPRM, and explain that the run-out phenomenon taking place on the milled surface through the irregular tool loci is induced by the spindle vibration under the intermittent cutting forces of UPRM. (iii) Since the particular spindle vibration is the periodical impulsive responses, it is interesting to note that the excitation frequency of the intermittent cutting forces, i.e. spindle speed, would influence the intermittent impulsive responses. (iv) Further, it is a potential means to improving the surface quality through the optimal selection of spindle speed. This will contribute significantly to the further improvement for
surface quality of ultra-precision raster milling, considering the effects of spindle vibration.
Acknowledgment This work was supported by The Research Grants Council of the Hong Kong Special Administrative Region of the People's Republic of China (Project No. PolyU 5287/10E). References [1] C.F. Cheung, W.B. Lee, S. To, A framework of a model-based simulation system for predicting surface generation in ultra-precision raster milling of freeform surfaces, in: Proceedings of American Society for Precision Engineering Annual Meeting, Orlando, Florida, October 24–29, 2004. [2] C.F. Cheung, L.B. Kong, W.B. Lee, S. To, Modelling and simulation of freeform surface generation in ultra-precision raster milling, Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture 220 (2006) 1787–1801. [3] M.N. Cheng, C.F. Cheung, W.B. Lee, S. To, L.B. Kong, Theoretical and experimental analysis of nano-surface generation in ultra-precision raster milling, International Journal of Machine Tools and Manufacture 48 (2008) 1090–1102. [4] L.B. Kong, Modeling of Ultra-precision Raster Milling and Characterization of Optical Freeform Surface. Ph.D. Dissertation, The Hong Kong Polytechnic University, Hong Kong, 2009. [5] M.N. Cheng, Optimization of Surface Generation in Ultra-Precision Multi-axis Raster Milling. M.Phil. Thesis, The Hong Kong Polytechnic University, Hong Kong, 2006. [6] Sujuan Wang, Modelling and Optimization of Cutting Strategy and Surface Generation in Ultra-Precision Raster Milling. Ph.D. Dissertation, The Hong Kong Polytechnic University, Hong Kong, 2010. [7] D.L. Martin, A.N. Tabenkin, F.G. Parsons, Precision spindle and bearing error analysis, International Journal of Machine Tools and Manufacture 35 (1995) 187–193. [8] E.R. Marsh, D.A. Arneson, M.J. Van Doren, S.A. Blystone, The effects of spindle dynamics on precision flycutting, in: Proceedings of American Society for Precision Engineering Annual Meeting, Norfolk, Virgina, October 09–14, 2005. [9] C.H. An, Y. Zhang, Q. Xu, F.H. Zhang, J.F. Zhang, L.J. Zhang, J.H. Wang, Modeling of dynamic characteristic of the aerostatic bearing spindle in an ultraprecision fly cutting machine, International Journal of Machine Tools and Manufacture 50 (2010) 374–385. [10] S.J. Zhang, S. To, C.F. Cheung, H.T. Wang, Dynamic characteristics of an aerostatic bearing spindle and its influence on surface topography in ultraprecision diamond turning, International Journal of Machine Tools and Manufacture 62 (2012) 1–12. [11] S.J. Zhang, S. To, H.T. Wang, A theoretical and experimental investigation into five-DOF dynamic characteristics of an aerostatic bearing spindle in ultraprecision diamond turning, International Journal of Machine Tools and Manufacture 71 (2013) 1–10.