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A generic instantaneous undeformed chip thickness model for the cutting force modeling in micromilling
Author’s Accepted Manuscript A generic instantaneous undeformed chip thickness model for the cutting force modeling in micromilling Li Kexuan, Zhu Kunpeng, Mei Tao www.elsevier.com/locate/ijmactool
PII: DOI: Reference:
S0890-6955(16)30011-6 http://dx.doi.org/10.1016/j.ijmachtools.2016.03.002 MTM3138
To appear in: International Journal of Machine Tools and Manufacture Received date: 27 October 2015 Revised date: 6 March 2016 Accepted date: 7 March 2016 Cite this article as: Li Kexuan, Zhu Kunpeng and Mei Tao, A generic instantaneous undeformed chip thickness model for the cutting force modeling in micromilling, International Journal of Machine Tools and Manufacture, http://dx.doi.org/10.1016/j.ijmachtools.2016.03.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A generic instantaneous undeformed chip thickness model for the cutting force modeling in micromilling Li Kexuan a,b, Zhu Kunpeng b,c,*, Mei Tao b a
Department of Precision Machinery and Precision Instrumentation, University of Science and Technology of China, Hefei, 230026, China. b
Institute of Advanced Manufacturing Technology, Hefei Institutes of Physical Science, Chinese Academy of Science, Huihong Building, Changwu Middle Road 801, Changzhou 213164, Jiangsu, China. c School of Logistics Engineering, Wuhan University of Technology, Heping Road 1178#, Wuhan 430063, Hubei,China.
Abstract The precise modeling of the instantaneous undeformed chip thickness is one of the key issues in the mechanics of micromilling. While most current models noticed the influences of the tool tip trochoidal trajectory and tool runout, they took account only the workpiece removed by immediate passing tooth but not more preceded teeth. These lead to inaccuracy when the single edge cutting occurred, which has been identified to be a prevalent phenomenon in micromilling operation. In this paper, the actual cutting area in micromilling is derived, and then a generic instantaneous undeformed chip thickness model is proposed by considering the cutting trajectory of all passing teeth in one cycle. Additionally, this study derives a criteria that could determine the single-edge-cutting phenomenon in multi-tooth micromilling from the geometric relations. The accuracy of the model is verified by the real experimental data and the result are shown superior to known models. Keywords: Micromilling, Chip Thickness, Tool Runout, Cutting Force
1. Introduction With the fast increasing demand of the miniature components and products in the fields such as aerospace, medical equipment and electronic communication device, the manufacturing methods of micro- and meso- scale parts become the research hotspot in these years [1-3]. Micromilling process, one of micro machining methods, has wide applications in micro and ultra-precision devices for its prominent capabilities in versatile material processing and complex 3D surface machining [4,5]. In addition, it can process high aspect ratios of parts [6], and has more excellent economic efficiency, and rapid process compared to micro electro-mechanical systems method [7]. Thus, micromilling technology has been one of the main machining technologies for complex micro parts. In general, Micromilling means the machining of the characteristic dimensions between 1 μm - 1 mm, and the milling cutters with the diameter below 1 mm [3]. Cutting force is one of the most fundamental and important aspect for understanding the mechanics of micromilling [8,9]. Meanwhile, the precise estimation of instantaneous undeformed chip thickness is the key to the modeling of cutting force
[10, 11]. Due to the large tool size and high feed rate in conventional milling, the instantaneous undeformed chip thickness often is calculated by considering the approximate circular trajectory of the tool tip [12]. However, with the dramatic decrease of tool diameter and feed rate, the calculation method of the instantaneous undeformed chip thickness in conventional milling is not suitable for micromilling, and any small error will cause the instantaneous undeformed chip thickness to change larger proportion in micromilling process. To meet this need, many researchers have modified the conventional instantaneous undeformed chip thickness model by considering the important influence factors in micromilling. Bao and Tansel [13] calculated the instantaneous undeformed chip thickness for two teeth micromilling operation by considering the trochoidal trajectory of the tool tip while the tool rotated and moved forward continuously. Tool runout was also considered in their later research [14] and the average error of maximum cutting force between the simulation by using the proposed model and experimental data was around 21%. Li et al [15] proposed a new instantaneous undeformed chip thickness algorithm for micromilling operation by considering the combination of an exact trochoidal trajectory of the tool tip and tool runout, and the average peak of cutting forces between computational by using the proposed model and experimental cutting forces was around 10%. Afazov et al [16] compared the changes of the instantaneous undeformed chip thickness between different tool runout lengths for two teeth micromilling operation by simulation, and discovered that the tool runout have very large effect on instantaneous undeformed chip thickness at low feed rate. Malekian et al [10] and Jun et al [17] introduced elastic recovery of the machined workpiece and dynamic characteristics of the micromilling tool to instantaneous undeformed chip thickness model, and obtained a precise prediction of cutting force in micromilling operation. Rodríguez and Labarga [18] established instantaneous undeformed chip thickness model for two teeth micromilling operation by considering tool runout, too deflection, size effect, and asymmetric cutting. Then the cutting forces were simulated, and showed satisfactory agreement with those data from micromilling experiments. Scholars have done a lot of research on calculation of the instantaneous undeformed chip thickness and achieved a lot of success, while most of studies were limited to two-tooth micromilling and many factors were neglected in determination of the instantaneous undeformed chip thickness. Among all the influence factors of the instantaneous undeformed chip thickness, tool runout is one of most important factors because of the ratio of tool runout to feed per tooth is much larger in micromilling. However, tool runout parameters are not easy to measure especially the tool runout angle. In micromilling, the cutter has a taper that connects the cutting edges of a smaller diameter to a shank of a larger diameter and this design introduces additional runout at the cutting edges. Therefore, the total runout can no longer be estimated at the tool shank and the values of tool runout parameters are more difficult to predict and control compared to macroscale tools [19]. In previous researches of micromilling, the tool runout parameters were immediate measured by dial indicator [16,18], capacitive sensor [20] or microscope [7,19,21]. Because of the small tool size, there were some measuring errors especially the runout angle in the methods. In conventional milling, some researches had used the methods that the analysis of the cutting force signals for determination of tool runout parameters. Wan et al [22] determined the tool runout parameters by the minimum error between the prediction and measurement of cutting force. Ko et al [23] selected the tool runout parameters that minimize the sum of the respective standard
deviations of the cutting force coefficients during one cutter revolution. In micromilling, the cutting edge radius of the cutter is comparable in size to the instantaneous undeformed chip thickness in micromilling and the minimum chip thickness becomes an important parameter. The specific energy will non-linear increase with a decrement in the instantaneous undeformed chip thickness when it is less than the minimum chip thickness [3,24]. Thus there would be some error if the traditional methods were used directly to determine the tool runout parameters. In this study, the real cutting areas are derived by including tool runout and a generic instantaneous undeformed chip thickness model is then proposed. Different from previous studies where only the immediate passing tooth is studied, the model considers the influence of the workpiece removed by all the previously passing teeth in one cycle. The influences of trochoidal trajectory of the tool tip and tool runout are also investigated. At the same time, a function for determining the single-edge-cutting phenomenon is derived. Then, the instantaneous undeformed chip thickness model generalizes the previous studies and an iterative algorithm is applied to estimate runout parameters with verifications from experiments.
2. The generic instantaneous undeformed chip thickness model of micromilling 2.1 The previous models and insufficiency The conventional instantaneous undeformed chip thickness model can be expressed in ideal conditions, i.e., in absence of trochoidal trajectory of the tool tip, tool runout and deflection, it can be approximately expressed in the formula (1). h( ) ft sin( ) (1) where h( ) is the instantaneous undeformed chip thickness(mm) at an instantaneous angle position (rad) of a milling tooth, and f t is the feed per tooth(mm/tooth). However, due to the rapid decreases of the tool diameter and feed rate, the conventional computation model of instantaneous undeformed chip thickness can’t precise describe the actual instantaneous undeformed chip thickness of micromilling. In the celebrated study, Bao and Tansel investigated the instantaneous undeformed chip thickness of micromilling operation [13,14]. They put forward the model based on the intersection between the current cutting edge’s trajectory and the following one’s. Li et al [15] improved the computational accuracy of Bao and Tansel’s model by iterative algorithm, while the results agreed with [13, 14]. The simulation of the model [14] is shown in Figure 1. It can be seen that tooth 1 can cut to the workpiece at only partially of the angular position (less than 180° in Figure 1(a)) or not entry of cutting (in Figure 1(b)), and tooth 2 can cut to the workpiece at very large angular position (more than 180° in Figure 1(a) or even up to 360° in Figure 1(b)). In addition, the maximum value of the instantaneous undeformed chip thickness of tooth 2 is more than two times of the feed per tooth when tool runout length exceeds a certain value.
1.5
tooth 1 tooth 2
h (m)
1
0.5
0 0
90
180 270 360 450 540 630 720 810 900
() (a) 0.5μm runout length
tooth 1 tooth 2
2
h (m)
1.5
1
0.5
0 0
90
180 270 360 450 540 630 720 810 900
()
(b) 1.2μm runout length Figure 1 Instantaneous undeformed chip thickness ( h ) simulate results according to the approach in [14]; simulation parameters: two teeth micromilling tool, 0.8mm tool diameter, 1μm/tooth feed per tooth and 60° tool runout angle Figure 2 is the trochoidal trajectory of tool tip of two teeth micromilling operation by considering tool runout. Area 1 and 2 is composed by the cutting trajectory of all previously passing teeth in one cycle. In the past researches, the cutting area of tooth 1 was composed by the sum of the area 1, 3 and 4, and the cutting area of tooth 2 is area 2. However, areas 3 and 4 have been removed by the previously passing teeth, and thus lacked accuracy. Additionally, the instantaneous undeformed chip thickness can’t be greater than the number of cutting edges multiplied by the feed per tooth. Based on previous analysis, a generic instantaneous undeformed chip thickness model is proposed in the following section.
The 1st cutting edge The 2nd cutting edge
ft
Tool diameter
3
2
1
4 Figure 2, Trochoidal trajectory of tool tip of the two teeth in micromilling operation
2.2 The generic instantaneous undeformed chip thickness model From the Figure 2 the actual instantaneous undeformed chip thicknes is the minimum distance of the intersections between the cutting trajectories of the current tooth and the previous teeth in one cycle, which connects the current cutting tip and the ideal cutter center. In this study, the instantaneous undeformed chip thickness is calculated by considering the effect of the exact trochoidal trajectory of tool tip, tool runout and the cutting trajectory of all previously passing teeth in one cycle. Tool runout often occurs in micromilling, and can be characterized by tool runout length ro and runout angle o as shown in Figure 3.
y
γ o
x
ro
Figure 3 The geometry of the tool runout The dotted and solid lines show the ideal and actual tool position respectively. The runout length ro indicates the deviation from the tool axis to the tool holder axis, and it is considered to be a constant along with the z-axis. The runout angle o is defined at the bottom section of the tool and measured anticlockwise from the offset direction to a given tooth tip. It is considered that the tool runout angle is changing along with the z-axis. The runout angle at z can be expressed in the equation (2).
o ( z )
(2)
where ( z ) is the radial lag angle at z, and ( z) z tan / r , r and are the tool radius and the helix angle of the micromilling tool respectively. Considering the effect of the tool runout, the trochoidal trajectory of the m -th tool tip can be written as [14]: ft 2 m x r sin(t ) ro sin(t ) (3) 60 M 2 m y r cos(t ) ro cos(t ) (4) M where f is the feed rate(mm/min), t is the time(s), m is the ordinal number of tool teeth and m 0,1...M 1 , the ordinal numbers of tool teeth are arranged 2 n counterclockwise, is the spindle circle speed (rad/s) and , n is the 60 spindle speed (rpm). Based on the equation (3) and (4), the trochoidal trajectory of the tool tip can be derived as: ft 2 m 2 m [ x ( ro sin(t ))]cos(t ) [ y ro cos(t )]sin(t ) 0 (5) 60 M M It is assumed that the current cutting edge tip at time tm , the previous k -th cutting edge tip at time tm k and the ideal cutter center is collinear. The line can be solved from equation (3)~(5) to the equation (6). f 2 m 2 k [ (tmk tm )]cos(tm ) r sin(tm k tm ) 60 M M (6) 2 m 2 m ro sin(tm k tm ) ro sin( ) 0 M
M
where k 1, 2...M . The equation (6) can be simplified: f 2 k ( )]cos( m ) r sin 2 n M 2 2 (m k ) 2 m ro sin[ ] ro sin( )0 M M
[
where m (1
(7)
4m ) t m M 2
m m t m t m k
k
2 k M
Because the real feed rate between the time tm and tm k is small relative to the tool radius, is a small value. Let sin , the value can be solved: kft cos ro r 2 m 2 (m k ) sin( ) o sin[ ] r r M r M (8) M cos ro 2 (m k ) 1 ft cos[ ] 2 r r M
where ft
f nM
m 2 The real feed rate is defined as: f M f f c (tm tmk ) kft t (9) 60 2 From the geometric relation, the actual cutting radiuses Rm and Rm k are affected by the tool runout [25]: Rm r 2 ro 2 2rro cos(
2 m ) M
(10)
2 (m k ) ] (11) M The instantaneous undeformed chip thickness calculation geometry can be shown in Figure 4. Om and Omk are the current cutter and the previous k -th cutter ideal center position, respectively. From the Figure 4: Rmk r 2 ro 2 2rro cos[
Rmk 2 H 2 fc2 2Hfc cos( m )
(12)
where H is the non-cutting edge length and can be solved from the above equation: H f c sin Rmk 2 ( fc cos )2
(13)
f c 2 cos 2 k 2 ft 2 cos2 f c4 cos 4 Considering , the non-cutting edge 1 and Rmk r r4 length H is simplified as:
ft M k 2 ft 2 cos2 H kft sin sin Rmk (14) 2 2r The distance of the intersections between the current cutting edge and the previous k -th cutting edge: f M k 2 f t 2 cos 2 hk Rm H kft sin t sin 2 2r (15) 2 m 2 ( m k ) r 2 ro 2 2rro cos( ) r 2 ro 2 2rro cos[ ] M M From Figure 2, the instantaneous undeformed chip thickness is the minimum value of hk , k 1, 2...M , and must be non-negative. The instantaneous undeformed chip thickness model can be written as: h max[0, min(hk )] (16)
h Om-k
fc
H
Rm
-k
Rm
m-k
m Om
Figure 4 Instantaneous undeformed chip thickness calculation geometry The formula (15) is a generalized representation of instantaneous undeformed chip thickness, and under particular conditions the model can be simplified to the previous studies. 1) If not consider the influence of the trochoidal trajectory of tool tip, tool runout and the cutting trajectory of previously passing teeth except the nearest one ( ft / r≈0 , ro 0 and 0 ), the equation (15) can be simplified as
h ft sin( ) which is the same as the instantaneous undeformed chip thickness model of the conventional milling in [12]. 2) If only consider the influence of trochoidal trajectory of the tool tip ( ro 0 and 0 ), and the cutter has two teeth ( M 2 ), the equation (15) can be written as the instantaneous undeformed chip thickness model of micromilling in [13]. 3) If not consider the influence of the cutting trajectory of the previously passing teeth except the nearest one ( k 1 ), and the cutter has two teeth ( M 2 ), the equation (15) can be written as the instantaneous undeformed chip thickness model of micromilling in [14]. 4) If not consider the influence of trochoidal trajectory of the tool tip ( ft / r≈0 and ro / r≈0 ) and assume ( ro 2 sin 2 ) is a small value, the equation (15) can be simplified: 2 m 2 (m k ) hk kft sin ro cos( ) ro cos[ ] (17) M M which is the same as the traditional instantaneous undeformed chip thickness model for example [22]. 5) And in the formula (17), when only accounting the nearest cutting trajectory ( k 1 ) and two teeth cutter ( M 2 ), the equation (17) can be further simplified as h ft sin (1)m1 2ro cos( )
which is the same as the instantaneous undeformed chip thickness model of Rodríguez and Labarga [18]. 2.3 Numerical Simulation of the Model
In order to compare with the previous instantaneous undeformed chip thickness models, the equation (16) is simulated in Figure 5.The tool runout angle is 120°, and the other simulation parameters of Figure 5(a) are the same as the Figure 1(a). Figure 5(b) is the simulation at 1.2μm runout length and 1.5μm runout length, and the other simulation parameters are the same as those of Figure 1(b). From the Figures, the simulation results of equation (16) have obvious difference with the previous models. According to the Figure 5(a), asymmetric cutting will occur when tool runout exists. When the tool runout length is lesser (0.5μm), the instantaneous undeformed chip thickness of tooth 1 is small, and cuts within a narrow range of cutting angle. The instantaneous undeformed chip thickness of tooth 2 is larger, and occurs within the scope of about 180 ° cutting angles. According to the Figure 5(b), the simulations at 1.2μm runout length and 1.5μm runout length are coincident. In addition, when the runout length exceeds a certain value (determination in the next section), there is only one tooth can cut the workpiece, and the instantaneous undeformed chip thickness of the tooth can't more than twice of the feed per tooth. In Figure 5(b), the instantaneous undeformed chip thickness of tooth 1 is zero, occurs single-edge-cutting phenomenon. And when the single-edge-cutting phenomenon occurs, the instantaneous undeformed chip thickness is only affected by the feed per tooth and don’t change when increasing tool runout length (e.g.1.2μm and 1.5μm). Simulation results show that the model agrees the actual situation of micromilling processes. 1.5
tooth 1 tooth 2
h (m)
1
0.5
0 0
90
180 270 360 450 540 630 720 810 900
()
(a) 0.5μm runout length
2
tooth 1 tooth 2
h (m)
1.5
1
0.5
0 0
90
180 270 360 450 540 630 720 810 900
() (b) 1.2μm and 1.5μm runout length (the simulations are coincident) Figure 5 Instantaneous undeformed chip thickness ( h ) simulate results according to the equation (16) tooth 1 tooth 2 tooth 3
h (m)
1.5
1
0.5
0 0
90
180 270 360 450 540 630 720 810 900
()
Figure 6 Instantaneous undeformed chip thickness in three teeth micromilling operation; simulation parameters: 0.8mm tool diameter, 1μm/tooth feed per tooth, 120° tool runout angle and 0.5μm runout length The equation (16) can be also used to more than two teeth micromilling process. Figure 6 is the simulation of three teeth micromilling process when tool runout occurs. Figure 6 shows that the tool runout has very important influence on the instantaneous undeformed chip thickness. The instantaneous undeformed chip thickness of tooth 1 and 2 don’t have intersection area, and tooth 3 is cutting the most part of the workpiece, which will shorten the tool life greatly. 2.4 Determination of single-edge-cutting phenomenon In micromilling, as the foregoing discussion, single-edge-cutting phenomenon occurs when the runout length exceeds a certain value. Due to sharp decreases of the tool diameter and feed rate, the single-edge-cutting phenomenon become a problem often appears in micromilling [18,26]. In single-edge-cutting process, only one tooth
can cut workpiece in different cutting periods, so the wear of the tooth involved in removal of workpiece is far greater than the other ones, as a result it reduces the life of tool, and this situation is to be avoided. The critical tool runout length can be determined from geometric relations, and the single-edge-cutting phenomenon can be forecasted. For M teeth micromilling processes, the maximum instantaneous undeformed chip thickness of each tooth is approximate occurred on the maximum x in a round. The trajectories of the feed direction of the current cutting tooth and the following k -th tooth are defined as xm and xm k , respectively, which can be written as:
ftm 2 m r sin(tm ) ro sin(tm ) 60 M ft 2 (m k ) xmk mk r sin[tmk ] ro sin(tmk ) 60 M where m 0...M and k 1...M 1 . xm
(18) (19)
From the equation (18), when (tm 2 m / M ) equals to ( 2 N / 2 ) ( N is an integer), xm reaches the maximum value in one cycle. Because is a small value and can be ignored in this section, the function mk can be written as:
mk xm xmk (20) 2 m 2 (m k ) ) ro cos[ ] M M Thus the determination functions of whether the cutting tooth is cutting workpiece or not is as following:
kft ro cos(
m min(mk )
(21)
Generally, the number of teeth that participate in cutting is the same as the number of m that are greater than zero. For all of the functions m 0 , all the teeth are involved in cutting. If only one of the functions m 0 , there will be only one tooth that can cut the workpiece, and under this condition the single-edge-cutting phenomenon occurs. As micromilling tools often have two teeth ( k 1 ), the equation (21) can be simplified as:
ft 2ro cos
(22)
When ft ≤ 2ro cos , single-edge-cutting phenomenon occurs, otherwise the two teeth cut alternately. For Figure 5(a), the function 0 , two teeth both participate in cutting. In the Figure 5(b), the function 0 , then the single-edge-cutting phenomenon occurs. 3. Determination of tool runout ( ro , o ) In the calculation of cutting force of micromilling operation, the micromilling tool can be divided into a finite number of disk elements and the total cutting force components acting on a tooth at a particular instant are obtained by numerically integrating the force components acting on an individual disk element. Finally, a
summation over all teeth engaged in cutting yields the total forces acting on the cutter at that time. Since the cutting forces in the axial direction are much little compared to the planar directions, only the forces in the X and Y directions have been considered in this study. The two orthogonal force components in Cartesian coordinates can be derived as follows from the expressions of the elemental tangential dFt and radial dFr forces as[23]: dFt K n hdz (23) dFr K f dFt where K n and K f are tangential and radial cutting force coefficients. Considering the geometric conditions, the integrated the cutting forces Fx and Fy can be expressed as:
Fx A11 F y A21
A12 K n A22 K f K n
(24)
M 1
r ex ( cos h)d m 0 tan en
where A11 M 1
M 1
M 1
r ex A12 ( sin h)d m 0 tan en
r ex A21 (sin h)d m 0 tan en r ex A22 ( cos h)d m 0 tan en
and en is the entry angle, ex is the exit angle. From the equation (24), it can be seen that the values of cutting force coefficients and instantaneous undeformed chip thickness are the major factors affecting the absolute values of cutting forces, and tool runout parameters are important factors to calculate them. In this study, the tool runout parameters include tool manufacturing and assembling error, and high speed rotation runout. In cutting operations, unit cutting force is proportional to the instantaneous undeformed chip thickness when the instantaneous undeformed chip thickness is much greater than the minimum chip thickness. However, in micromilling, due to the decrease of the instantaneous undeformed chip thickness, cutting process can be divided into three stages: shearing dominant, ploughing dominant and transition stages. In the stages of ploughing dominant and transition, a non-linear increase of the unit cutting force with a decrement of the instantaneous undeformed chip thickness. In this study, an iterative algorithm is put forward to determine tool runout parameters with eliminating the influence of minimum chip thickness shown in Fig. 7 ( ro max is the maximum possible of the tool runout length, hmin is the minimum chip thickness).
ro 0, o 0 Sampling angles h( ) hmin h( ) hmin
Experimental force date
&
Angles ro 0.001
o 1o
Calculation of Kn ( ), K f ()
Calculate the average value Kn ( ), K f ( )
Calculate the squared difference (ro , o ) between the simulated and measured cutting forces at the angles
ro≤ro max
o≤360o
Selection of ro , o having minimum (ro , o )
Figure 7 Algorithm for determining tool runout parameters in micromilling
The detailed steps of this iterative algorithm are as follows: 1) A length of cutting force signal is extracted, and then averaged to a revolution according to the set of sampling angles. Sampling angles are set arbitrarily by sampling rate f s (Hz) and spindle speed n , and they have the equal interval angle . The measured cutting force data can be synchronized by sampling the identical range with the simulated Fy . 2) The tool runout parameters are substituted iteratively into the equation (16), starting from certain initial values ( ro 0 , o 0o ). The angles are contained
by the sampling angles from step 1, and they are set to satisfy the instantaneous undeformed chip thickness h exceeding the minimum chip thickness along the cutting teeth. 3) The revolution of cutting force signal is substituted on the left hand side of equation (24), and then K n ( ) and K f ( ) are obtained from the equation at the angles using Cramer’s Rule [27]. The cutting force coefficients K n ( ) and
K f ( ) are averaged as K n and K f respectively. 4) The cutting forces are simulated by using the values of K n and K f from step 3. Then the squared difference (ro , o ) between the simulated and measured cutting forces at the angles is estimated. 5) The values of ro and o that produce the minimum (ro , o ) are selected to be the real cutter runout.
4. Experimental validation 4.1 Experimental setup A series of tests were conducted on a high-precision vertical milling machine driven by a 22kw spindle drive motor, and the spindle speed variation is between 5,000-30,000 rpm. The micromilling machine is shown in Fig. 8. The cutting force was measured with a Kistler 9256A 3-channel piezoelectric dynamometer under the workpiece. The cutting force output was recorded on a Sony digital tape recorder. The sampling rate of the cutting forces was 6,000 Hz. The workpiece materials were pure copper. The micro tools used in this study were 500 μm diameter micromilling tools with the helix and shank taper angles of approximately 30 and 16 degrees, respectively. The phenomenon of self-excited vibrations would distort the cutting forces a great deal, thus the cutting speed were carefully examined to stay within a suitable range to avoid self-excited vibrations and keep the reasonable level of productivity. The appropriate feed rate was selected, so sufficient and effective sampling angles were used to eliminate the influence of the minimum chip thickness on the determination of tool runout parameters. The cutting conditions of two tests were listed in table 1.
Spindle system
Micromilling tool Dynamometer
Table
Workpiece Figure 8 Experimental setup
Table 1. Cutting conditions Tool diameter: 0.5 mm Workpiece material: Copper Spindle speed: 18,000 rpm Milling method: up-milling Test No. Depth of cut (µm) Radial depth of cut (μm) 1 2
150 120
Helix angle: 30° Two-teeth cutter Feed rate (mm/min)
225 300
150 150
4.2 Result and discussion Tool edge radius of the micromilling tool was measured to be 2μm with an optical microscope shown in Fig. 9. Generally, the minimum chip thickness was 14-43% of the tool edge radius for different cutting variables and workpiece material in micromilling [28]. It was 25% of the tool edge radius for copper material according to [29]. Considering the transition stage, in this study the undeformed chip thickness was set as 1.2μm ( hmin in the Figure 7). It was set up to 1.5 to 2 times of the minimum chip thickness because of the frequent transitions from plowing-dominant to shear-dominant stages; this was also discussed in [20].
Fig. 9. Tool edge radius of micromilling cutter.
The tool runout parameters and cutting force coefficients were obtained by the iterative algorithm in section 3. In order to avoid accidental factors such as material uneven, a number of the cutting force signals were extracted, and then averaged to a revolution according to the set of sampling angles after eliminating the drift and lag of the cutting force signals. The tool runout parameters, cutting force coefficients and sampling angles estimated from tests, are given in Table 2. Table 2. Estimated cutting force coefficients, tool runout parameters and sampling angles. Test no.
Kn (N mm-2)
Kf
ro (mm)
γo (deg.)
Φ(deg.)
1
8514.2
-1.6058
0.003
76
{0,18…360}
2
12501
-0.8278
0.003
30
{0,18…360}
The comparisons of two tests (test 1 and 2) between the simulated and experimental cutting force (time-domain and frequency-domain) in the global coordinate systems (X and Y) are given in Figure 10 and 11, respectively. As shown in the figures, the profiles both in time and frequency domains of the cutting forces show good agreement between the experiments and the simulations. The prediction error of maximum cutting forces between the simulation and experimental data is shown in Table 3. The prediction errors of using the Bao and Tansel's model [14] are also listed below for a comparison, with which both cutting force coefficients determined by this study and by [14] are applied. As can be seen from Table 3, the prediction of cutting force by using the proposed model in this paper shows better accuracy than the model in [14], especially for Test 2. When comparing the accuracy by their respective coefficients, the Test 1 also shows much improvement of prediction accuracy, with error decreasing from 12.40% to 3.69%. It is noted that the mechanistic cutting force models decided jointly by instantaneous undeformed chip thickness and the cutting force coefficients, and cutting force coefficients were determined by the cutting force experimental data and the instantaneous undeformed chip thickness model. If the cutting force coefficients were obtained only through the basic principles of materials and metal cutting theory [30], there would be large prediction errors of the cutting force. Table 3. The prediction error of maximum cutting force between the simulation and experimental data. Error of maximum cutting force
The proposed model in this paper
Bao and Tansel [14]: cutting force coefficients from [14]
Bao and Tansel[14]: cutting force coefficients from Table 2
Test 1
Test 2
Test 1
Test 2
Test 1
Test 2
△Fx
3.55%
4.01%
6.07%
3.27%
3.55%
11.85%
△Fy
3.69%
12.24%
12.40%
14.52%
3.70%
26.52%
From the cutting force signal of the tests, test 1 is multiple teeth alternating cut, and test 2 occurs single-edge-cutting phenomenon. Test 1 and 2 have the same tool runout length, but different tool runout angles, which can explain that the tool runout angle has great influence on cutting force. In micromilling, tool runout angle is more
difficult to control because of the taper, and it became an important problem in micromilling. To decide the cutting conditions of test 1 and 2 by the formula (22), functions 1 0 (test 1: multiple teeth alternating cutting) and 2 0 (test 2: single-edge-cutting in all depth of cut) are consistent with the experimental results. The single-edge-cutting phenomenon can be also found from the spectrogram. In test 1 and 2, tooth passing frequency is 600Hz ( 2n / 60 ). The amplitude of 300Hz (the half of the tooth passing frequency) is small in Figure 10, while the value is much larger due to single-edge-cutting phenomenon in Figure 11. In Figure 10(a), the simulated and experimental cutting force has a larger error near the (360N+198°) because of the size effect, which illustrates the ploughing force became a larger proportion in the total cutting force when the instantaneous undeformed chip thickness is small. Meanwhile, there is part of the prediction error in around (360N+108°) may due to the frictional force between the bottom edge and workpiece [31], which would be further investigated in the future study.
Fx experimental Fx theoretical
6
Fx (N)
4
2
0
0
180
360
540
720
900 1080 1260 1440 1620 1800
()
FFT-Fx experimental FFT-Fx theoretical
2
FFT-Fx
1.5
1
0.5
0 0
500
1000
1500
2000
Frequency (Hz)
(a) Feed direction
2500
3000
Fy experimental Fy theoretical
10
Fy (N)
8 6 4 2 0 0
180
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Figure 11 Comparison of cutting forces and the frequency spectrums between theoretical and experimental at test 2 In this study, the prediction of cutting forces shows satisfactory agreement with the experiments by applying the generic instantaneous undeformed chip thickness model. The model considers the influences of trochoidal trajectory of the tool tip, tool runout and the workpiece removed by all the previously passing teeth. Single-edge-cutting phenomenon has important effect on tool wear and machining accuracy, and it can be determined by a function. If the function is less than zero, single-edge-cutting phenomenon occurs, and the installation angle of the cutter can be adjusted before micromilling operations, and the tool life can be increased. The tool runout parameters could be determined by a simple method which will provide information for the accurate prediction of cutting force.
5. Conclusions A precise model of the instantaneous undeformed chip thickness is key to the modeling of cutting force and would enhance the understanding of the micromilling mechanics as a result. While in micromilling the tool runout and the single-edge-cutting phenomenon bring difficulties in accurately modeling of the instantaneous undeformed chip thickness. This study addresses these issues by introducing a generic instantaneous undeformed chip thickness model which includes runout factors, and derives a simple function that can determine the single-edge-cutting phenomenon. The model takes account the entire cutting trajectory in one full cycle in determining the chip thickness, and the model is more reasonable and accurate than previous models that the real chip thickness was calculated only by immediate passing tooth. The model has shown good generalizations and can be simplified to the existed models under specific conditions. It has been verified by experiments and shown to have better accuracy compared to previous celebrated model. In the future research, the model will be extended to include the parameters that characterize the size effect and tool deflection factors. Acknowledgement: The authors would like to thank the National University of Singapore for the permission of using these data for the research. References [1] [2]
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Highlights: 1.
Introduced a generic instantaneous undeformed chip thickness model in micromilling.
2.
Derived a function that determine the single-edge-cutting phenomenon.
3.
Showed good generalizations to the existed models.
4.
Showed better accuracy compared to previous celebrated model.
PII: DOI: Reference:
S0890-6955(16)30011-6 http://dx.doi.org/10.1016/j.ijmachtools.2016.03.002 MTM3138
To appear in: International Journal of Machine Tools and Manufacture Received date: 27 October 2015 Revised date: 6 March 2016 Accepted date: 7 March 2016 Cite this article as: Li Kexuan, Zhu Kunpeng and Mei Tao, A generic instantaneous undeformed chip thickness model for the cutting force modeling in micromilling, International Journal of Machine Tools and Manufacture, http://dx.doi.org/10.1016/j.ijmachtools.2016.03.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A generic instantaneous undeformed chip thickness model for the cutting force modeling in micromilling Li Kexuan a,b, Zhu Kunpeng b,c,*, Mei Tao b a
Department of Precision Machinery and Precision Instrumentation, University of Science and Technology of China, Hefei, 230026, China. b
Institute of Advanced Manufacturing Technology, Hefei Institutes of Physical Science, Chinese Academy of Science, Huihong Building, Changwu Middle Road 801, Changzhou 213164, Jiangsu, China. c School of Logistics Engineering, Wuhan University of Technology, Heping Road 1178#, Wuhan 430063, Hubei,China.
Abstract The precise modeling of the instantaneous undeformed chip thickness is one of the key issues in the mechanics of micromilling. While most current models noticed the influences of the tool tip trochoidal trajectory and tool runout, they took account only the workpiece removed by immediate passing tooth but not more preceded teeth. These lead to inaccuracy when the single edge cutting occurred, which has been identified to be a prevalent phenomenon in micromilling operation. In this paper, the actual cutting area in micromilling is derived, and then a generic instantaneous undeformed chip thickness model is proposed by considering the cutting trajectory of all passing teeth in one cycle. Additionally, this study derives a criteria that could determine the single-edge-cutting phenomenon in multi-tooth micromilling from the geometric relations. The accuracy of the model is verified by the real experimental data and the result are shown superior to known models. Keywords: Micromilling, Chip Thickness, Tool Runout, Cutting Force
1. Introduction With the fast increasing demand of the miniature components and products in the fields such as aerospace, medical equipment and electronic communication device, the manufacturing methods of micro- and meso- scale parts become the research hotspot in these years [1-3]. Micromilling process, one of micro machining methods, has wide applications in micro and ultra-precision devices for its prominent capabilities in versatile material processing and complex 3D surface machining [4,5]. In addition, it can process high aspect ratios of parts [6], and has more excellent economic efficiency, and rapid process compared to micro electro-mechanical systems method [7]. Thus, micromilling technology has been one of the main machining technologies for complex micro parts. In general, Micromilling means the machining of the characteristic dimensions between 1 μm - 1 mm, and the milling cutters with the diameter below 1 mm [3]. Cutting force is one of the most fundamental and important aspect for understanding the mechanics of micromilling [8,9]. Meanwhile, the precise estimation of instantaneous undeformed chip thickness is the key to the modeling of cutting force
[10, 11]. Due to the large tool size and high feed rate in conventional milling, the instantaneous undeformed chip thickness often is calculated by considering the approximate circular trajectory of the tool tip [12]. However, with the dramatic decrease of tool diameter and feed rate, the calculation method of the instantaneous undeformed chip thickness in conventional milling is not suitable for micromilling, and any small error will cause the instantaneous undeformed chip thickness to change larger proportion in micromilling process. To meet this need, many researchers have modified the conventional instantaneous undeformed chip thickness model by considering the important influence factors in micromilling. Bao and Tansel [13] calculated the instantaneous undeformed chip thickness for two teeth micromilling operation by considering the trochoidal trajectory of the tool tip while the tool rotated and moved forward continuously. Tool runout was also considered in their later research [14] and the average error of maximum cutting force between the simulation by using the proposed model and experimental data was around 21%. Li et al [15] proposed a new instantaneous undeformed chip thickness algorithm for micromilling operation by considering the combination of an exact trochoidal trajectory of the tool tip and tool runout, and the average peak of cutting forces between computational by using the proposed model and experimental cutting forces was around 10%. Afazov et al [16] compared the changes of the instantaneous undeformed chip thickness between different tool runout lengths for two teeth micromilling operation by simulation, and discovered that the tool runout have very large effect on instantaneous undeformed chip thickness at low feed rate. Malekian et al [10] and Jun et al [17] introduced elastic recovery of the machined workpiece and dynamic characteristics of the micromilling tool to instantaneous undeformed chip thickness model, and obtained a precise prediction of cutting force in micromilling operation. Rodríguez and Labarga [18] established instantaneous undeformed chip thickness model for two teeth micromilling operation by considering tool runout, too deflection, size effect, and asymmetric cutting. Then the cutting forces were simulated, and showed satisfactory agreement with those data from micromilling experiments. Scholars have done a lot of research on calculation of the instantaneous undeformed chip thickness and achieved a lot of success, while most of studies were limited to two-tooth micromilling and many factors were neglected in determination of the instantaneous undeformed chip thickness. Among all the influence factors of the instantaneous undeformed chip thickness, tool runout is one of most important factors because of the ratio of tool runout to feed per tooth is much larger in micromilling. However, tool runout parameters are not easy to measure especially the tool runout angle. In micromilling, the cutter has a taper that connects the cutting edges of a smaller diameter to a shank of a larger diameter and this design introduces additional runout at the cutting edges. Therefore, the total runout can no longer be estimated at the tool shank and the values of tool runout parameters are more difficult to predict and control compared to macroscale tools [19]. In previous researches of micromilling, the tool runout parameters were immediate measured by dial indicator [16,18], capacitive sensor [20] or microscope [7,19,21]. Because of the small tool size, there were some measuring errors especially the runout angle in the methods. In conventional milling, some researches had used the methods that the analysis of the cutting force signals for determination of tool runout parameters. Wan et al [22] determined the tool runout parameters by the minimum error between the prediction and measurement of cutting force. Ko et al [23] selected the tool runout parameters that minimize the sum of the respective standard
deviations of the cutting force coefficients during one cutter revolution. In micromilling, the cutting edge radius of the cutter is comparable in size to the instantaneous undeformed chip thickness in micromilling and the minimum chip thickness becomes an important parameter. The specific energy will non-linear increase with a decrement in the instantaneous undeformed chip thickness when it is less than the minimum chip thickness [3,24]. Thus there would be some error if the traditional methods were used directly to determine the tool runout parameters. In this study, the real cutting areas are derived by including tool runout and a generic instantaneous undeformed chip thickness model is then proposed. Different from previous studies where only the immediate passing tooth is studied, the model considers the influence of the workpiece removed by all the previously passing teeth in one cycle. The influences of trochoidal trajectory of the tool tip and tool runout are also investigated. At the same time, a function for determining the single-edge-cutting phenomenon is derived. Then, the instantaneous undeformed chip thickness model generalizes the previous studies and an iterative algorithm is applied to estimate runout parameters with verifications from experiments.
2. The generic instantaneous undeformed chip thickness model of micromilling 2.1 The previous models and insufficiency The conventional instantaneous undeformed chip thickness model can be expressed in ideal conditions, i.e., in absence of trochoidal trajectory of the tool tip, tool runout and deflection, it can be approximately expressed in the formula (1). h( ) ft sin( ) (1) where h( ) is the instantaneous undeformed chip thickness(mm) at an instantaneous angle position (rad) of a milling tooth, and f t is the feed per tooth(mm/tooth). However, due to the rapid decreases of the tool diameter and feed rate, the conventional computation model of instantaneous undeformed chip thickness can’t precise describe the actual instantaneous undeformed chip thickness of micromilling. In the celebrated study, Bao and Tansel investigated the instantaneous undeformed chip thickness of micromilling operation [13,14]. They put forward the model based on the intersection between the current cutting edge’s trajectory and the following one’s. Li et al [15] improved the computational accuracy of Bao and Tansel’s model by iterative algorithm, while the results agreed with [13, 14]. The simulation of the model [14] is shown in Figure 1. It can be seen that tooth 1 can cut to the workpiece at only partially of the angular position (less than 180° in Figure 1(a)) or not entry of cutting (in Figure 1(b)), and tooth 2 can cut to the workpiece at very large angular position (more than 180° in Figure 1(a) or even up to 360° in Figure 1(b)). In addition, the maximum value of the instantaneous undeformed chip thickness of tooth 2 is more than two times of the feed per tooth when tool runout length exceeds a certain value.
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(b) 1.2μm runout length Figure 1 Instantaneous undeformed chip thickness ( h ) simulate results according to the approach in [14]; simulation parameters: two teeth micromilling tool, 0.8mm tool diameter, 1μm/tooth feed per tooth and 60° tool runout angle Figure 2 is the trochoidal trajectory of tool tip of two teeth micromilling operation by considering tool runout. Area 1 and 2 is composed by the cutting trajectory of all previously passing teeth in one cycle. In the past researches, the cutting area of tooth 1 was composed by the sum of the area 1, 3 and 4, and the cutting area of tooth 2 is area 2. However, areas 3 and 4 have been removed by the previously passing teeth, and thus lacked accuracy. Additionally, the instantaneous undeformed chip thickness can’t be greater than the number of cutting edges multiplied by the feed per tooth. Based on previous analysis, a generic instantaneous undeformed chip thickness model is proposed in the following section.
The 1st cutting edge The 2nd cutting edge
ft
Tool diameter
3
2
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4 Figure 2, Trochoidal trajectory of tool tip of the two teeth in micromilling operation
2.2 The generic instantaneous undeformed chip thickness model From the Figure 2 the actual instantaneous undeformed chip thicknes is the minimum distance of the intersections between the cutting trajectories of the current tooth and the previous teeth in one cycle, which connects the current cutting tip and the ideal cutter center. In this study, the instantaneous undeformed chip thickness is calculated by considering the effect of the exact trochoidal trajectory of tool tip, tool runout and the cutting trajectory of all previously passing teeth in one cycle. Tool runout often occurs in micromilling, and can be characterized by tool runout length ro and runout angle o as shown in Figure 3.
y
γ o
x
ro
Figure 3 The geometry of the tool runout The dotted and solid lines show the ideal and actual tool position respectively. The runout length ro indicates the deviation from the tool axis to the tool holder axis, and it is considered to be a constant along with the z-axis. The runout angle o is defined at the bottom section of the tool and measured anticlockwise from the offset direction to a given tooth tip. It is considered that the tool runout angle is changing along with the z-axis. The runout angle at z can be expressed in the equation (2).
o ( z )
(2)
where ( z ) is the radial lag angle at z, and ( z) z tan / r , r and are the tool radius and the helix angle of the micromilling tool respectively. Considering the effect of the tool runout, the trochoidal trajectory of the m -th tool tip can be written as [14]: ft 2 m x r sin(t ) ro sin(t ) (3) 60 M 2 m y r cos(t ) ro cos(t ) (4) M where f is the feed rate(mm/min), t is the time(s), m is the ordinal number of tool teeth and m 0,1...M 1 , the ordinal numbers of tool teeth are arranged 2 n counterclockwise, is the spindle circle speed (rad/s) and , n is the 60 spindle speed (rpm). Based on the equation (3) and (4), the trochoidal trajectory of the tool tip can be derived as: ft 2 m 2 m [ x ( ro sin(t ))]cos(t ) [ y ro cos(t )]sin(t ) 0 (5) 60 M M It is assumed that the current cutting edge tip at time tm , the previous k -th cutting edge tip at time tm k and the ideal cutter center is collinear. The line can be solved from equation (3)~(5) to the equation (6). f 2 m 2 k [ (tmk tm )]cos(tm ) r sin(tm k tm ) 60 M M (6) 2 m 2 m ro sin(tm k tm ) ro sin( ) 0 M
M
where k 1, 2...M . The equation (6) can be simplified: f 2 k ( )]cos( m ) r sin 2 n M 2 2 (m k ) 2 m ro sin[ ] ro sin( )0 M M
[
where m (1
(7)
4m ) t m M 2
m m t m t m k
k
2 k M
Because the real feed rate between the time tm and tm k is small relative to the tool radius, is a small value. Let sin , the value can be solved: kft cos ro r 2 m 2 (m k ) sin( ) o sin[ ] r r M r M (8) M cos ro 2 (m k ) 1 ft cos[ ] 2 r r M
where ft
f nM
m 2 The real feed rate is defined as: f M f f c (tm tmk ) kft t (9) 60 2 From the geometric relation, the actual cutting radiuses Rm and Rm k are affected by the tool runout [25]: Rm r 2 ro 2 2rro cos(
2 m ) M
(10)
2 (m k ) ] (11) M The instantaneous undeformed chip thickness calculation geometry can be shown in Figure 4. Om and Omk are the current cutter and the previous k -th cutter ideal center position, respectively. From the Figure 4: Rmk r 2 ro 2 2rro cos[
Rmk 2 H 2 fc2 2Hfc cos( m )
(12)
where H is the non-cutting edge length and can be solved from the above equation: H f c sin Rmk 2 ( fc cos )2
(13)
f c 2 cos 2 k 2 ft 2 cos2 f c4 cos 4 Considering , the non-cutting edge 1 and Rmk r r4 length H is simplified as:
ft M k 2 ft 2 cos2 H kft sin sin Rmk (14) 2 2r The distance of the intersections between the current cutting edge and the previous k -th cutting edge: f M k 2 f t 2 cos 2 hk Rm H kft sin t sin 2 2r (15) 2 m 2 ( m k ) r 2 ro 2 2rro cos( ) r 2 ro 2 2rro cos[ ] M M From Figure 2, the instantaneous undeformed chip thickness is the minimum value of hk , k 1, 2...M , and must be non-negative. The instantaneous undeformed chip thickness model can be written as: h max[0, min(hk )] (16)
h Om-k
fc
H
Rm
-k
Rm
m-k
m Om
Figure 4 Instantaneous undeformed chip thickness calculation geometry The formula (15) is a generalized representation of instantaneous undeformed chip thickness, and under particular conditions the model can be simplified to the previous studies. 1) If not consider the influence of the trochoidal trajectory of tool tip, tool runout and the cutting trajectory of previously passing teeth except the nearest one ( ft / r≈0 , ro 0 and 0 ), the equation (15) can be simplified as
h ft sin( ) which is the same as the instantaneous undeformed chip thickness model of the conventional milling in [12]. 2) If only consider the influence of trochoidal trajectory of the tool tip ( ro 0 and 0 ), and the cutter has two teeth ( M 2 ), the equation (15) can be written as the instantaneous undeformed chip thickness model of micromilling in [13]. 3) If not consider the influence of the cutting trajectory of the previously passing teeth except the nearest one ( k 1 ), and the cutter has two teeth ( M 2 ), the equation (15) can be written as the instantaneous undeformed chip thickness model of micromilling in [14]. 4) If not consider the influence of trochoidal trajectory of the tool tip ( ft / r≈0 and ro / r≈0 ) and assume ( ro 2 sin 2 ) is a small value, the equation (15) can be simplified: 2 m 2 (m k ) hk kft sin ro cos( ) ro cos[ ] (17) M M which is the same as the traditional instantaneous undeformed chip thickness model for example [22]. 5) And in the formula (17), when only accounting the nearest cutting trajectory ( k 1 ) and two teeth cutter ( M 2 ), the equation (17) can be further simplified as h ft sin (1)m1 2ro cos( )
which is the same as the instantaneous undeformed chip thickness model of Rodríguez and Labarga [18]. 2.3 Numerical Simulation of the Model
In order to compare with the previous instantaneous undeformed chip thickness models, the equation (16) is simulated in Figure 5.The tool runout angle is 120°, and the other simulation parameters of Figure 5(a) are the same as the Figure 1(a). Figure 5(b) is the simulation at 1.2μm runout length and 1.5μm runout length, and the other simulation parameters are the same as those of Figure 1(b). From the Figures, the simulation results of equation (16) have obvious difference with the previous models. According to the Figure 5(a), asymmetric cutting will occur when tool runout exists. When the tool runout length is lesser (0.5μm), the instantaneous undeformed chip thickness of tooth 1 is small, and cuts within a narrow range of cutting angle. The instantaneous undeformed chip thickness of tooth 2 is larger, and occurs within the scope of about 180 ° cutting angles. According to the Figure 5(b), the simulations at 1.2μm runout length and 1.5μm runout length are coincident. In addition, when the runout length exceeds a certain value (determination in the next section), there is only one tooth can cut the workpiece, and the instantaneous undeformed chip thickness of the tooth can't more than twice of the feed per tooth. In Figure 5(b), the instantaneous undeformed chip thickness of tooth 1 is zero, occurs single-edge-cutting phenomenon. And when the single-edge-cutting phenomenon occurs, the instantaneous undeformed chip thickness is only affected by the feed per tooth and don’t change when increasing tool runout length (e.g.1.2μm and 1.5μm). Simulation results show that the model agrees the actual situation of micromilling processes. 1.5
tooth 1 tooth 2
h (m)
1
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0 0
90
180 270 360 450 540 630 720 810 900
()
(a) 0.5μm runout length
2
tooth 1 tooth 2
h (m)
1.5
1
0.5
0 0
90
180 270 360 450 540 630 720 810 900
() (b) 1.2μm and 1.5μm runout length (the simulations are coincident) Figure 5 Instantaneous undeformed chip thickness ( h ) simulate results according to the equation (16) tooth 1 tooth 2 tooth 3
h (m)
1.5
1
0.5
0 0
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180 270 360 450 540 630 720 810 900
()
Figure 6 Instantaneous undeformed chip thickness in three teeth micromilling operation; simulation parameters: 0.8mm tool diameter, 1μm/tooth feed per tooth, 120° tool runout angle and 0.5μm runout length The equation (16) can be also used to more than two teeth micromilling process. Figure 6 is the simulation of three teeth micromilling process when tool runout occurs. Figure 6 shows that the tool runout has very important influence on the instantaneous undeformed chip thickness. The instantaneous undeformed chip thickness of tooth 1 and 2 don’t have intersection area, and tooth 3 is cutting the most part of the workpiece, which will shorten the tool life greatly. 2.4 Determination of single-edge-cutting phenomenon In micromilling, as the foregoing discussion, single-edge-cutting phenomenon occurs when the runout length exceeds a certain value. Due to sharp decreases of the tool diameter and feed rate, the single-edge-cutting phenomenon become a problem often appears in micromilling [18,26]. In single-edge-cutting process, only one tooth
can cut workpiece in different cutting periods, so the wear of the tooth involved in removal of workpiece is far greater than the other ones, as a result it reduces the life of tool, and this situation is to be avoided. The critical tool runout length can be determined from geometric relations, and the single-edge-cutting phenomenon can be forecasted. For M teeth micromilling processes, the maximum instantaneous undeformed chip thickness of each tooth is approximate occurred on the maximum x in a round. The trajectories of the feed direction of the current cutting tooth and the following k -th tooth are defined as xm and xm k , respectively, which can be written as:
ftm 2 m r sin(tm ) ro sin(tm ) 60 M ft 2 (m k ) xmk mk r sin[tmk ] ro sin(tmk ) 60 M where m 0...M and k 1...M 1 . xm
(18) (19)
From the equation (18), when (tm 2 m / M ) equals to ( 2 N / 2 ) ( N is an integer), xm reaches the maximum value in one cycle. Because is a small value and can be ignored in this section, the function mk can be written as:
mk xm xmk (20) 2 m 2 (m k ) ) ro cos[ ] M M Thus the determination functions of whether the cutting tooth is cutting workpiece or not is as following:
kft ro cos(
m min(mk )
(21)
Generally, the number of teeth that participate in cutting is the same as the number of m that are greater than zero. For all of the functions m 0 , all the teeth are involved in cutting. If only one of the functions m 0 , there will be only one tooth that can cut the workpiece, and under this condition the single-edge-cutting phenomenon occurs. As micromilling tools often have two teeth ( k 1 ), the equation (21) can be simplified as:
ft 2ro cos
(22)
When ft ≤ 2ro cos , single-edge-cutting phenomenon occurs, otherwise the two teeth cut alternately. For Figure 5(a), the function 0 , two teeth both participate in cutting. In the Figure 5(b), the function 0 , then the single-edge-cutting phenomenon occurs. 3. Determination of tool runout ( ro , o ) In the calculation of cutting force of micromilling operation, the micromilling tool can be divided into a finite number of disk elements and the total cutting force components acting on a tooth at a particular instant are obtained by numerically integrating the force components acting on an individual disk element. Finally, a
summation over all teeth engaged in cutting yields the total forces acting on the cutter at that time. Since the cutting forces in the axial direction are much little compared to the planar directions, only the forces in the X and Y directions have been considered in this study. The two orthogonal force components in Cartesian coordinates can be derived as follows from the expressions of the elemental tangential dFt and radial dFr forces as[23]: dFt K n hdz (23) dFr K f dFt where K n and K f are tangential and radial cutting force coefficients. Considering the geometric conditions, the integrated the cutting forces Fx and Fy can be expressed as:
Fx A11 F y A21
A12 K n A22 K f K n
(24)
M 1
r ex ( cos h)d m 0 tan en
where A11 M 1
M 1
M 1
r ex A12 ( sin h)d m 0 tan en
r ex A21 (sin h)d m 0 tan en r ex A22 ( cos h)d m 0 tan en
and en is the entry angle, ex is the exit angle. From the equation (24), it can be seen that the values of cutting force coefficients and instantaneous undeformed chip thickness are the major factors affecting the absolute values of cutting forces, and tool runout parameters are important factors to calculate them. In this study, the tool runout parameters include tool manufacturing and assembling error, and high speed rotation runout. In cutting operations, unit cutting force is proportional to the instantaneous undeformed chip thickness when the instantaneous undeformed chip thickness is much greater than the minimum chip thickness. However, in micromilling, due to the decrease of the instantaneous undeformed chip thickness, cutting process can be divided into three stages: shearing dominant, ploughing dominant and transition stages. In the stages of ploughing dominant and transition, a non-linear increase of the unit cutting force with a decrement of the instantaneous undeformed chip thickness. In this study, an iterative algorithm is put forward to determine tool runout parameters with eliminating the influence of minimum chip thickness shown in Fig. 7 ( ro max is the maximum possible of the tool runout length, hmin is the minimum chip thickness).
ro 0, o 0 Sampling angles h( ) hmin h( ) hmin
Experimental force date
&
Angles ro 0.001
o 1o
Calculation of Kn ( ), K f ()
Calculate the average value Kn ( ), K f ( )
Calculate the squared difference (ro , o ) between the simulated and measured cutting forces at the angles
ro≤ro max
o≤360o
Selection of ro , o having minimum (ro , o )
Figure 7 Algorithm for determining tool runout parameters in micromilling
The detailed steps of this iterative algorithm are as follows: 1) A length of cutting force signal is extracted, and then averaged to a revolution according to the set of sampling angles. Sampling angles are set arbitrarily by sampling rate f s (Hz) and spindle speed n , and they have the equal interval angle . The measured cutting force data can be synchronized by sampling the identical range with the simulated Fy . 2) The tool runout parameters are substituted iteratively into the equation (16), starting from certain initial values ( ro 0 , o 0o ). The angles are contained
by the sampling angles from step 1, and they are set to satisfy the instantaneous undeformed chip thickness h exceeding the minimum chip thickness along the cutting teeth. 3) The revolution of cutting force signal is substituted on the left hand side of equation (24), and then K n ( ) and K f ( ) are obtained from the equation at the angles using Cramer’s Rule [27]. The cutting force coefficients K n ( ) and
K f ( ) are averaged as K n and K f respectively. 4) The cutting forces are simulated by using the values of K n and K f from step 3. Then the squared difference (ro , o ) between the simulated and measured cutting forces at the angles is estimated. 5) The values of ro and o that produce the minimum (ro , o ) are selected to be the real cutter runout.
4. Experimental validation 4.1 Experimental setup A series of tests were conducted on a high-precision vertical milling machine driven by a 22kw spindle drive motor, and the spindle speed variation is between 5,000-30,000 rpm. The micromilling machine is shown in Fig. 8. The cutting force was measured with a Kistler 9256A 3-channel piezoelectric dynamometer under the workpiece. The cutting force output was recorded on a Sony digital tape recorder. The sampling rate of the cutting forces was 6,000 Hz. The workpiece materials were pure copper. The micro tools used in this study were 500 μm diameter micromilling tools with the helix and shank taper angles of approximately 30 and 16 degrees, respectively. The phenomenon of self-excited vibrations would distort the cutting forces a great deal, thus the cutting speed were carefully examined to stay within a suitable range to avoid self-excited vibrations and keep the reasonable level of productivity. The appropriate feed rate was selected, so sufficient and effective sampling angles were used to eliminate the influence of the minimum chip thickness on the determination of tool runout parameters. The cutting conditions of two tests were listed in table 1.
Spindle system
Micromilling tool Dynamometer
Table
Workpiece Figure 8 Experimental setup
Table 1. Cutting conditions Tool diameter: 0.5 mm Workpiece material: Copper Spindle speed: 18,000 rpm Milling method: up-milling Test No. Depth of cut (µm) Radial depth of cut (μm) 1 2
150 120
Helix angle: 30° Two-teeth cutter Feed rate (mm/min)
225 300
150 150
4.2 Result and discussion Tool edge radius of the micromilling tool was measured to be 2μm with an optical microscope shown in Fig. 9. Generally, the minimum chip thickness was 14-43% of the tool edge radius for different cutting variables and workpiece material in micromilling [28]. It was 25% of the tool edge radius for copper material according to [29]. Considering the transition stage, in this study the undeformed chip thickness was set as 1.2μm ( hmin in the Figure 7). It was set up to 1.5 to 2 times of the minimum chip thickness because of the frequent transitions from plowing-dominant to shear-dominant stages; this was also discussed in [20].
Fig. 9. Tool edge radius of micromilling cutter.
The tool runout parameters and cutting force coefficients were obtained by the iterative algorithm in section 3. In order to avoid accidental factors such as material uneven, a number of the cutting force signals were extracted, and then averaged to a revolution according to the set of sampling angles after eliminating the drift and lag of the cutting force signals. The tool runout parameters, cutting force coefficients and sampling angles estimated from tests, are given in Table 2. Table 2. Estimated cutting force coefficients, tool runout parameters and sampling angles. Test no.
Kn (N mm-2)
Kf
ro (mm)
γo (deg.)
Φ(deg.)
1
8514.2
-1.6058
0.003
76
{0,18…360}
2
12501
-0.8278
0.003
30
{0,18…360}
The comparisons of two tests (test 1 and 2) between the simulated and experimental cutting force (time-domain and frequency-domain) in the global coordinate systems (X and Y) are given in Figure 10 and 11, respectively. As shown in the figures, the profiles both in time and frequency domains of the cutting forces show good agreement between the experiments and the simulations. The prediction error of maximum cutting forces between the simulation and experimental data is shown in Table 3. The prediction errors of using the Bao and Tansel's model [14] are also listed below for a comparison, with which both cutting force coefficients determined by this study and by [14] are applied. As can be seen from Table 3, the prediction of cutting force by using the proposed model in this paper shows better accuracy than the model in [14], especially for Test 2. When comparing the accuracy by their respective coefficients, the Test 1 also shows much improvement of prediction accuracy, with error decreasing from 12.40% to 3.69%. It is noted that the mechanistic cutting force models decided jointly by instantaneous undeformed chip thickness and the cutting force coefficients, and cutting force coefficients were determined by the cutting force experimental data and the instantaneous undeformed chip thickness model. If the cutting force coefficients were obtained only through the basic principles of materials and metal cutting theory [30], there would be large prediction errors of the cutting force. Table 3. The prediction error of maximum cutting force between the simulation and experimental data. Error of maximum cutting force
The proposed model in this paper
Bao and Tansel [14]: cutting force coefficients from [14]
Bao and Tansel[14]: cutting force coefficients from Table 2
Test 1
Test 2
Test 1
Test 2
Test 1
Test 2
△Fx
3.55%
4.01%
6.07%
3.27%
3.55%
11.85%
△Fy
3.69%
12.24%
12.40%
14.52%
3.70%
26.52%
From the cutting force signal of the tests, test 1 is multiple teeth alternating cut, and test 2 occurs single-edge-cutting phenomenon. Test 1 and 2 have the same tool runout length, but different tool runout angles, which can explain that the tool runout angle has great influence on cutting force. In micromilling, tool runout angle is more
difficult to control because of the taper, and it became an important problem in micromilling. To decide the cutting conditions of test 1 and 2 by the formula (22), functions 1 0 (test 1: multiple teeth alternating cutting) and 2 0 (test 2: single-edge-cutting in all depth of cut) are consistent with the experimental results. The single-edge-cutting phenomenon can be also found from the spectrogram. In test 1 and 2, tooth passing frequency is 600Hz ( 2n / 60 ). The amplitude of 300Hz (the half of the tooth passing frequency) is small in Figure 10, while the value is much larger due to single-edge-cutting phenomenon in Figure 11. In Figure 10(a), the simulated and experimental cutting force has a larger error near the (360N+198°) because of the size effect, which illustrates the ploughing force became a larger proportion in the total cutting force when the instantaneous undeformed chip thickness is small. Meanwhile, there is part of the prediction error in around (360N+108°) may due to the frictional force between the bottom edge and workpiece [31], which would be further investigated in the future study.
Fx experimental Fx theoretical
6
Fx (N)
4
2
0
0
180
360
540
720
900 1080 1260 1440 1620 1800
()
FFT-Fx experimental FFT-Fx theoretical
2
FFT-Fx
1.5
1
0.5
0 0
500
1000
1500
2000
Frequency (Hz)
(a) Feed direction
2500
3000
Fy experimental Fy theoretical
10
Fy (N)
8 6 4 2 0 0
180
360
540
720
900 1080 1260 1440 1620 1800
()
FFT-Fy experimental FFT-Fy theoretical
5
FFT-Fy
4
3
2
1
0 0
500
1000
1500
2000
2500
3000
Frequency (Hz)
(b) Normal direction Figure 10 Comparison of cutting forces and the frequency spectrums between theoretical and experimental at test 1
Fx experimental Fx theoretical
10 8
Fx (N)
6 4 2 0 -2 0
180
360
540
720
900 1080 1260 1440 1620 1800
()
2
FFT-Fx experimental FFT-Fx theoretical
FFT-Fx
1.5
1
0.5
0 0
500
1000
1500
2000
Frequency (Hz)
(a) Feed direction
2500
3000
Fy experimental Fy theoretical
15
Fy (N)
12 9 6 3 0 0
180
360
540
720
900 1080 1260 1440 1620 1800
()
5
FFT-Fy experimental FFT-Fy theoretical
FFT-Fy
4
3
2
1
0 0
500
1000
1500
2000
2500
3000
Frequency (Hz) (b) Normal direction
Figure 11 Comparison of cutting forces and the frequency spectrums between theoretical and experimental at test 2 In this study, the prediction of cutting forces shows satisfactory agreement with the experiments by applying the generic instantaneous undeformed chip thickness model. The model considers the influences of trochoidal trajectory of the tool tip, tool runout and the workpiece removed by all the previously passing teeth. Single-edge-cutting phenomenon has important effect on tool wear and machining accuracy, and it can be determined by a function. If the function is less than zero, single-edge-cutting phenomenon occurs, and the installation angle of the cutter can be adjusted before micromilling operations, and the tool life can be increased. The tool runout parameters could be determined by a simple method which will provide information for the accurate prediction of cutting force.
5. Conclusions A precise model of the instantaneous undeformed chip thickness is key to the modeling of cutting force and would enhance the understanding of the micromilling mechanics as a result. While in micromilling the tool runout and the single-edge-cutting phenomenon bring difficulties in accurately modeling of the instantaneous undeformed chip thickness. This study addresses these issues by introducing a generic instantaneous undeformed chip thickness model which includes runout factors, and derives a simple function that can determine the single-edge-cutting phenomenon. The model takes account the entire cutting trajectory in one full cycle in determining the chip thickness, and the model is more reasonable and accurate than previous models that the real chip thickness was calculated only by immediate passing tooth. The model has shown good generalizations and can be simplified to the existed models under specific conditions. It has been verified by experiments and shown to have better accuracy compared to previous celebrated model. In the future research, the model will be extended to include the parameters that characterize the size effect and tool deflection factors. Acknowledgement: The authors would like to thank the National University of Singapore for the permission of using these data for the research. References [1] [2]
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Highlights: 1.
Introduced a generic instantaneous undeformed chip thickness model in micromilling.
2.
Derived a function that determine the single-edge-cutting phenomenon.
3.
Showed good generalizations to the existed models.
4.
Showed better accuracy compared to previous celebrated model.