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Prediction of chatter stability in high-speed finishing end milling considering multi-mode dynamics
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 2585–2591
journal homepage: www.elsevier.com/locate/jmatprotec
Prediction of chatter stability in high-speed finishing end milling considering multi-mode dynamics W.X. Tang a,∗ , Q.H. Song a , S.Q. Yu b , S.S. Sun a , B.B. Li a , B. Du a , X. Ai a a b
School of Mechanical Engineering, Shandong University, Jinan, Shandong 250061, PR China School of Software Engineering, Beijing University of Technology, Beijing, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history:
The strong demand for increasing productivity and workpiece quality in high-speed milling
Received 26 July 2007
make the machine-tool system has to operate close to the limit of its dynamic stability.
Received in revised form
This requires that the chatter stability is predicted accurately to determine the optimal
14 May 2008
milling parameters. An analytical stability prediction method was proposed with multi-
Accepted 7 June 2008
degree-of-freedom (MDOF) system modal analysis. This paper describes the development of this new method which allows considering the effects of multi-mode dynamics of system, higher excited frequency (i.e. tooth passing frequency) and wider spindle speed range on
Keywords:
stability limits in high-speed milling, and these to help in selection of milling parameters
Stability prediction
for a maximum material removal rates (MRR) in real operations without chatter. Some tests
High-speed milling
were carried out to demonstrate the quality of this method used in real machining. Final,
Chatter
the main influencing factors of stability limits in high-speed milling were analyzed. © 2008 Elsevier B.V. All rights reserved.
Multi-mode dynamics Frequency response function
1.
Introduction
High-speed milling is growing very rapidly in aerospace, automotive, die making and many other industries due to its advantages such as higher material removal rates (MRR), better surface finish and lower cost and so on. However, the unstable milling due to chatter vibration is not only one of the main limitations for productivity and workpiece quality, especially for a finishing high-speed milling, but also shorten the life of the machines and tools evidently. In particular, the aerospace, etc. industries have a strong demand for increasing MRR. Therefore, machine-tool system has to operate close to the limit of its dynamic stability. In addition, the manufactured structural parts in the aerospace industry become larger and more complex. If chatter vibration occurs during milling, the scrap rate of produced workpieces rises. This leads to immense economic losses. To run the milling process close
∗
to the limit between stable and unstable cut, the prediction of the stability limit is absolutely necessary. However, the predicting of stability in high-speed milling is more complicated than the conventional machining due to the rotating tool resulting in a time varying dynamics, periodical cutting forces and chip-load directions, and multi-degree-of-freedom (MDOF) structural dynamics. Thus, the investigation for predicting stability of high-speed milling against chatter vibration is vital for the goals of producing high quality products and maximum MRR in real machining. A wide range of research has been conduced to predict the stability with different methods, including analytical prediction method (Altintas and Budak, 1995; Budak, 2006), stability boundary analysis with the radial basis neural network (RBNN) (Yoon and Kim, 2007), experimental prediction method, such as using audio signal measurements, etc. (Weingaertner et al., 2006), and analyticalexperimental prediction method for stability lobes (Solis et al.,
Corresponding author. Tel.: +86 531 8839 5617; fax: +86 531 8839 6708. E-mail address: [email protected] (W.X. Tang). 0924-0136/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2008.06.003
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2004), etc. Deshayes (2007) analyzed an equivalent tool face for the cutting speed range prediction of complex grooved tools. Gagnol et al. (2007) elaborated a dynamic model of a highspeed spindle-bearing system on the basis of rotor dynamics predictions. Movahhedy and Mosaddegh (2006) developed the dynamics model of machine-spindle-tool-workpiece system considering the gyroscopic effects of spindle and used to predict the stability limits in high-speed milling. Mane et al. (2008) recently presented a stability-based spindle speed control method during flexible workpiece high-speed milling. These researches played an important part in practical manufactures. The stability of milling process is dependent on the dynamic behavior of machine-tool-workpiece system, which is often expressed in terms of the frequency response function (FRF) of system at the tool tip. The FRF of system at the tool tip is usually determined by considering the contribution of first or lower order mode characteristics of system only. At lower milling speeds, the effects of these higher order mode’s characteristics of system on the stability can be safely ignored. However, with the rising speed of the spindle, the increase of tool teeth pass frequency leads to the higher order modal frequencies of system are exited. The effects of multi-mode dynamics characteristics of MDOF system become more pronounced and the process stability deviate from the case expressed only by first or lower order modes of system. This may leads to a wrong prediction of the regions of a stable cutting process, especially at high-speed milling. It is therefore worthwhile to investigate as to how the stability during high-speed milling is crucially affected by the multi-mode characteristics of MDOF system and accurately predicted. In this paper, a prediction method of stability in highspeed finishing end milling is proposed with respects to the multi-mode dynamics characteristics of MDOF system using the mode synthesis technique. The proposed method was expected to provide practical guidance to machine-tool user for optimal process planning of spindle speeds and depth of cuts in milling operations. To verify stability lobes obtained by the proposed method, some milling tests were conducted. Several main influence factors of stability in high-speed milling were analyzed.
2.
Stability analysis for end milling
2.1.
Dynamics model and milling force
In order to compute the FRF at the tool tip, which is required for chatter stability analysis, a cross-sectional view at the contact zone of end tool and workpiece, which are included as part of multi-degree-of-freedom system, in high-speed milling is shown in Fig. 1 using the model presented by Altintas and Budak (1995). The analyzed tool with the diameter D and the number z of equally spaced teeth rotates at a constant angular velocity ˝ (rad/s). The radial immersion angle of the ith tooth varies with time as: ϕi (t) = ˝t + 2(i − 1)/z. The machine-toolworkpiece system is excited by the milling force F(t) at the tool and workpiece contact zone causing dynamic displacement
Fig. 1 – The cross-sectional view of an end mill in milling process.
q(t) of the structure governed by the following equation ¨ + C q(t) ˙ + Kq(t) = F(t) M q(t)
(1)
where M, C and K are the mass, damping, stiffness matrices of high-speed milling system respectively, which all are matrices of n × n orders (n = 1, 2, 3, . . .). For a point on the ith cutting tooth, the tangential (Ft,i ) and radial (Fr,i ) milling force components are assumed proportional to the chip-load defined by the chip thickness hi (t) and the axial depth of cut ap Ft,i (t) = Kt ap hi (t) Fr,i (t) = Kr Ft,i (t)
(2)
where Kt and Kr denote the special cutting force coefficients in the tangential and radial directions, respectively. And the tangential and radial forces can be resolved in the feed (x) and normal (y) directions Fx,i (t) = ıi (t)[−Ft,i (t) cos ϕi (t) − Fr,i (t) sin ϕi (t)] Fy,i (t) = ıi (t)[Ft,i (t) sin ϕi (t) − Fr,i (t) cos ϕi (t)]
(3)
where denoting a unit step function determining whether or not the ith tooth is cutting by ıi (t). The instantaneous chip thickness hi (t) consists of a static component due to the feed motion with feed per tooth ft , hs,i (t) = ft sin ϕi (t), and a dynamic component hd,i (t) which is generated due to the displacement of tool qtd,i (t) and the displacement of workpiece qwd,i (t). Because only the dynamic component hd,i (t) contributes to the dynamic chip-load regeneration mechanism, so hi (t) can be expressed as hi (t)
= hd,i (t) = qtd,i (t) − qwd,i (t) = ıi (t)[x(t, T) sin ϕi (t) + y(t, T) cos ϕi (t)]
(4)
where x(t, T) = [xt (t) − xw (t)] − [xt (t − T) − xw (t − T)] and y(t, T) = [yt (t) − yw (t)] − [yt (t − T) − yw (t − T)] describe the surface regeneration, i.e. the difference between the tool positions
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at the present and previous tooth passes; T = 2/(z˝) is the tooth passing period. And xt (t), xw (t), yt (t) and yw (t) present displacements of the tool and workpiece in x and y directions, respectively. Substituting hi (t) into Eqs. (2) and (3), the dynamic milling forces can be resolved in x and y directions and reduced as {F(t)} = 12 ap Kt [A0 ]{(t)}
(5)
where
z [A0 (t)] = 2
⎧ ⎪ A0xx = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪A = ⎪ ⎨ 0xy
1 2
A0xx A0yx
A0xy A0yy
cos 2ϕ − 2
(6)
Kr ϕ + sin 2ϕ Kt
{F(r) }eiωc t =
˛en ˛ex ˛en
1 Kr −2ϕ − sin 2ϕ + cos 2ϕ 2 Kt
˛ex
˛en
⎪ ⎪ A0yx ⎪ ⎪ ⎪ ˛ex ⎪ ⎪ ˛en ⎪ ⎪ ⎩ A0yy = 1 − cos 2ϕ − 2 Kr ϕ − Kr sin 2ϕ 1 Kr = 2ϕ − sin 2ϕ + cos 2ϕ 2 Kt Kt
2
Kt
(7)
det I −
(Kr − ωr Mr )˚r = 0 ˚Tr M˚r = I
(8)
Suppose the modes of system are not dense, and the eigenvalues can be arranged as 0 < ω1 < ω2 < · · · < ωr < ωr+1 < · · · < ωn
(9)
Based on the orthogonality of mode characteristics of MDOF system (Kelly, 1996), the dynamics equation of the rth order mode can be written as follows (r = 1, 2, · · ·, n)
= R + I = −
j=1
=0
(14)
n j=1,j = / r
[G(j) (iω)]
(11)
z ap Kt (1 − e−iωc T ) 4
(15)
2R (1 + ε2 ) zKt
(16)
sin ωc T I = R 1 − cos ωc T
(17)
where ε=
The above can be solved simply by finding the spindle rotating speed N and the tooth passing period T: N=
60 zT
T=
1 ( + 2k) ωc
(18) (k = 0, 1, . . .)
(19)
where = − 2cot−1 ε is the phase shift between the inner and outer modulations (present and previous teeth). Therefore, for given cutting geometry, cutting force coefficients, eigenvalue and chatter frequency, the corresponding stability limit axial depth of cut and spindle speed can be determine from Eqs. (16) and (18).
2.3. [G(j) (iω)] = [G(r) (iω)] +
where R and I are the real and imaginary parts of eigenvalue , respectively. And the final expression for chatter free axial depth of cut is given as
(10)
where Mr , Cr , Kr , Fr (t) and qr (t) are the modal mass, modal damping, modal stiffness matrices, modal forces vector and modal space vector (xr (t) = r qr (t)) of the rth order mode of system in the generalized co-ordinate, respectively. And the frequency response function of the n-DOF system can be expressed as
[G(iω)] =
1 (r) ap Kt (1 − e−iωc T ) G0 (iω) 2
and the eigenvalue of the characteristic equation is
ap = −
n
(13)
Eq. (12) has a non-trivial solution if its determinant is zero
˛ex
Chatter stability
Mr q¨ r (t) + Cr q˙ r (t) + Kr qr (t) = F r (t)
(12)
where {F(r) } represents the amplitude of the dynamic milling force {Fr (t)} at the tool- workpiece contact zone corresponding to the rth mode of the system. And assumed (r)
The characteristic equation of the multi-degree-of-freedom system expressed by Eq. (1) has n eigenvalues, on the other words, n natural frequencies (ω1 , ω2 , . . ., ωr , . . ., ωn ) and n eigenvectors ({1 }, {2 }, . . ., {r }, . . ., {n }). The rth eigenvalue ωr and eigenvector r satisfy the following eigen-equations and orthogonality conditions
1 ap Kt (1 − e−iωc T )[A0 ][G(r) (iω)]{F(r) }eiωc t 2
[G0 (iω)] = [A0 ][G(r) (iω)]
here, ˛en and ˛ex are entry angle and exit angle of the tool, respectively.
2.2.
where matrix [G(r) (iω)] which is the FRF of the rth order mode is defined between the tool and workpiece. The last item of above equation is the residue item except the rth order FRF and can be eliminated when the vibration occur at a dominate frequency ω which is around the rth order modal frequency ωnr of system based on above analysis. The vibrations are assumed to occur at a chatter frequency ωc , when a critical axial depth of cut is taken. The forces {Fr (t)} are described as a harmonic functions and the closedloop equation of the milling operation in the generalized co-ordinates and the Eq. (10) can be expressed in frequency domain as the following form (Tang, 2005)
Stability limits analysis
Based on the orthogonality of modal characteristics of MDOF system (Kelly, 1996), when vibration occurs at a chatter fre-
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quency ωc , which is around the rth order modal frequency ωnr , in high-speed milling process, the eigenvalue in Eq. (15) satisfies orthogonality condition Eqs. (8) and (9) and can be expressed as (Altintas and Budak, 1995): =
1
(20)
(r) [G0 (iω)] (r)
And [G0 (iω)] can be also expressed in the following form (r)
(r)
(r)
[G0 (iω)] = Re[G0 (iω)] + Im[G0 (iω)] (r)
(21)
(r)
where, Re[G0 (iω)]) and Im[G0 (iω)] are real and imaginary components of the FRF corresponding the rth order mode of the system. The FRF implicitly contains the mode characteristics, i.e. modal frequency (ωnr ), modal damping ratio ( r ) and modal stiffness (kr ), of the system. Once these characteristics are obtained analytically or experimentally, the both components can be determined as follows 1 − 2r
(r)
Re[G0 (iω)] = A0xx
2
−2i r r
(r)
Im[G0 (iω)] = A0xx
2
kr [(1 − 2r ) + (2 r r ) ]
2
2
kr [(1 − 2r ) + (2 r r ) ]
(22a)
(22b) Fig. 2 – Flow chart of obtaining stability lobes.
where r = ω/ωnr = f/fnr ; f = ω/2 is the excited frequency. By taking the average number za of teeth during the cut za =
zae 2D
(23)
where ae is the radial depth of cut. And the value of ae is small during a finishing end milling process and ae and ft both vary (r) within a narrow range. Then substituting za and [G0 (iω)] into Eqs. (20) and (16), the stable critical axial depth of cut is given by (r)
alim = −
1
(24)
(r)
2Kt za Re[G0 (iω)]
The equation shows that the stability limit of milling may (r) exist no other than the real components Re[G0 (iω)] of the FRF (r)
are less than zero, because of alim > 0 in practical milling processes. The differentiation about the parameter r of Eq. (22a) (r)
leads to the minimum of Re[G0 (iω)] when the condition r = 1/2 exists (1 + 2 r ) (r)
Re[G0 (iω)]min = −
1 [4kr r (1 + r )]
(25)
The minimum of the stability limits coordinating to all modes can be expressed as (r)
amin = min[amin ]
(r = 1, 2, · · ·, n)
(27)
The amin can be explained that the milling is stable at any spindle speeds as long as the axial depth of milling is less than the amin .
2.4.
Creating stability lobes diagram during milling
Based on the above analysis, the stability limits of high-speed milling processes can be predicted with respect to multiple mode characteristics of MDOF systems. To apply effectively the proposed method to predict the stability limits in real high-speed milling, the flow chart and process of creating stability lobes diagram using the proposed method are given and shown in Figs. 2 and 3, respectively: (1) Obtain the dynamic characteristics and milling processes parameters (fnr , mr , kr , r , Kt , Kr , ft , ae , z, za ) of the machinetool-workpiece system using experimental analysis; (2) Select one excite frequency (f) surrounding the rth order modal frequency (fnr ) of the system; (r)
So the minimum of the stability limits corresponding the rth order mode of the system can be written as
(r)
(3) Calculate the real (Re[G0 (iω)]) and imaginary (Im[G0 (iω)]) component of the frequency response function, Eqs. (22a) and (22b); (r)
(r)
amin =
4kr r (1 + r ) 2Kt za
(26)
(4) Calculate the stability limit (alim ) of milling, Eq. (26); (5) Calculate the spindle rotating speed (N) from Eq. (18) for each stability lobe k = 0, 1, 2, . . ., Eq. (19);
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Fig. 4 – Stability lobes obtained by predicting simulation.
Fig. 3 – Process of obtaining stability lobes.
(6) Repeat from Step (3) by scanning the excite frequencies; and, (7) Repeat from Step (2) by scanning the next order mode (fnr+1 ). To obtain the stability lobes of high-speed milling with respect to multi-mode characteristics of MDOF system, it is necessary to set an appropriate step load (f = fr+1 − fr ) and apply the above method repeatedly to all modes of FRF. And as a practical example, a stability lobes diagram, which based upon the modal parameters of a five-axis machining center (DMU-70V) with the solid carbide tool (diameter is 10 mm), is obtained by the proposed method and shown in Fig. 4. It should be noticed that the horizontal axis in Fig. 4 is the product of spindle speed (N) and number of tool teeth (z).
3.
Experimental validation
To validate the proposed stability predicting method, a series of experiments were performed on a five-axis machining center DMU-70V whose maximum spindle speed is 18,000 rpm. Three solid carbide end mills with one, two and four teeth (z = 1, 2, 4, D = 10 mm) were selected. The workpiece material was 0.45% C (24 HRC) steel. The modes characteristics of the machining center configuration are identified using the experimental modal analysis technique.
The test system is shown in Fig. 5. In Fig. 5, the symbols ux and uy denote the displacement responses measured by non-contact displacement sensors in feed (x) and normal (y) directions respectively. The ax and ay denote the acceleration responses measured by acceleration sensors. And fx , fy and fz denote the milling forces measured by a Kistler dynamometer in x, y and z directions respectively. In the milling tests, the selection of spindle rotating speeds (N) from 5000 to 17,000 rpm with the increments of 2,000 rpm was intended to verify the calculated stability limits for finishing end milling processes with different axial depths of cut (ap ). For each spindle speed, ap started with 2 mm and it was increased with the steps of 2 mm in each new cutting passing. Since the primary aim of the tests was to analyze and verify the stability of the finishing milling process where low immersions used usually in practice, the radial depth of cut (ae were selected as 0.3, 0.5 and 1 mm and the feed per tooth (ft ) was 0.1 mm/tooth in up- and down-milling. The effect of the ft on stability is small and can be ignored (Kivanc and Budak, 2004). The results of tests with three solid carbide tools (z = 1, 2 and 4) are shown in Figs. 6–8, and compared with the calculated results by the proposed method, respectively. The tag ‘’ represents a milling process without chatter and ‘×’ represents an unstable milling process with chatter. The curves represent the calculated critical stability limits. In Figs. 6–8, it can be observed that measured and calculated stability limits showed reasonable agreement and the configurations of stability lobes during milling process using three tools with different number (z) of tool teeth, z = 1, 2, 4, were clearly different even under same milling conditions, i.e. N, ap , ae and ft .
Fig. 5 – Testing system.
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4. Influencing factors of the stability in high-speed finishing end milling In order to obtain perfect surface finish, milling process has to be stable. The aim of predicting stability is to determine the optimum milling process parameters, i.e. N, ap , ae and ft , for a maximum material removal rate (MRR) without chatter. The MRR is proportional to the multiplication of milling parameters and the number of tool teeth z MRR = ap ae Nzft
Fig. 6 – Stability lobes for z = 1, ae = 0.3 mm, ft = 0.1 mm/tooth, up-milling.
(28)
In general, in a finishing end milling process, value of both ae and ft and their variable range are small. For simplicity, effects of both ae and ft on stability can be neglected. The detail of considering effects of both axial and radial depths of cut can be found in reference (Budak, 2006). Therefore, the rest of the analysis focuses on the mainly influencing factors except both ae and ft and their optimization to maximize the stability limits against chatter.
4.1.
Frequency response functions
The frequency response function implicitly contains the mode characteristics of the system, like the natural frequency (fnr ), modal stiffness (kr ) and modal damping ratio ( r ). If the mode characterisics of the system change, there is a direct consequence for the process stability. The optimum spindle speed ranges, in which a large depth of cut can be achieved without chatter, will also shift. If the modal stiffness or/and damping ratio are different, the minimum (amin ) of stabile axial depth of cut (ap ), which depended on the modal parameters closely, is also different. According to the Eqs. (22a) and (24), the larger are the modal stiffness (kr ) and modal damping ratio ( r ), the smaller is the (r)
Fig. 7 – Stability lobes for z = 2, ae = 0.3 mm, ft = 0.1 mm/tooth, up-milling.
absolute value of real component Re[G0 (iω)] of FRF, thus, the larger stability limit axial depth of cut (alim ) can be acquired. For example, the calculated and measured stability lobes both show that the ranges of spindle speed where the largest cutting depths can be achieved without chatter are different in Figs. 6–8, respectively (in Fig. 6: N = 11 krpm, alim = 14 mm; in Fig. 7: N = 16 krpm, alim = 14.5 mm; and in Fig. 8: N = 15 krpm, alim = 14.5 mm). The maximum MRR obtained during stable milling (N = 11 krpm, ap = 14 mm, z = 1) shown in Fig. 6 is only approximate 1/5 of the milling (N = 15 krpm, ap = 14 mm, z = 4) shown in Fig. 8. In practice, it is expected that machine-tool system operate at a range of spindle speed close to its permitted maximum speed without chatter for the aim of enhancing both the finish quality and MRR. Hence, optimizing the modal stiffness and damping ratio of higher order modes, which corresponding higher spindle speed zone, is more significant than other lower order modes in high-speed milling.
4.2. Fig. 8 – Stability lobes for z = 4, ae = 0.3 mm, ft = 0.1 mm/tooth, up-milling.
Modal frequency
In Fig. 3, it is observed that each stability lobe was located along abscissa (spindle speed N) in stability lobes diagram depended on the magnitude of corresponding a certain
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 2585–2591
order mode’s natural frequency. For the same machine-toolworkpiece system, the stability lobes corresponding larger value of the modal frequencies are located at righter of the abscissa, i.e. higher spindle speed zone. This higher speed zone should be optimized emphatically for maximum MRR in real high-speed milling. Comparing between Figs. 6–8, it can be found that the maximum modal frequencies (fnr ) excited in available spindle speed range (0–18000 rpm) are respectively the first order fn1 in Fig. 6, the fourth order fn4 in Fig. 7 and the fifth order fn5 in Fig. 8 on a same machine-tool-workpiece system. The critical stable depth of cut (alim ) corresponding these different modal frequencies is also different. This may lead to an evident different MRR which can be achieved without chatter. Therefore, the actual multi-mode dynamics of the investigated machine-tool-workpiece system has to be considered in highspeed milling to obtain the improvement of the stability limits predicting accuracy and optimum milling process parameters.
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multi-mode dynamics of MFOD system, higher excited frequency (i.e. tooth passing frequency) and wider spindle speed range on stability limits in high-speed milling. The process and flow chart of predicting stability limits using the proposed method were presented for applying this method into the real milling operations. And the stability lobes diagrams were obtained using this method and validated by milling experiments. The use of the proposed method in prediction of stability contributes to obtain an improvement of the stability lobes accuracy and optimum milling conditions for enhancing the surface finish quality and MRR in practice. The results of calculated and experimented show that prediction of the stability limits, especially under higher spindle speed, required considering the actual multi-mode dynamics of the investigated machine-tool-workpieces system. Otherwise, unexpected chatter vibration may occur which will damage the workpiece, the tool, or the machine-tool.
Acknowledgements 4.3.
The number (z) of tool teeth
Since the tooth passing frequency is proportional to the number (z) of tool teeth, a tool with larger number (z) of teeth can be selected to increase the MRR. Simultaneous the higher order modal frequencies of system may also be excited in lower spindle speed milling. Although the difference of the number (z) of teeth hardly affects the dynamics of the whole system and the minimum of the stability limits, it may lead to the change of the location of every lobe at abscissa (N) in stability lobes diagram. Therefore, when the tools with different number (z) of teeth are used at the same milling condition, the configurations of stability lobes may appear obvious different. In order to assess the effects of different number (z) of teeth on the stability limits in high-speed milling, the stability lobes between three tools with different number (z) of teeth, z = 1, 2 and 4, are compared in Figs. 6–8. It can be found that the configurations of stability lobes, the maximum stable ap , N and MRR obtained in three cases using different tools (z = 1, 2, 4) were different evidently. A trend of increase of MRR with increasing number (z) of teeth at same milling parameters condition is existent, however, to obtain a maximum MRR without chatter, the optimum milling parameters must be determined using the stability lobes curve predicted accurately. Therefore, the importance of selecting number (z) of teeth is two folds. First, in some real cases, axial depth ap is fixed due to the geometry of the part, special for the finishing milling with low radial immersion, thus the maximum stable N and z must be determined. Second, the available maximum spindle speed N for a certain machine-tool is fixed, the maximum MRR can only be achieved by optimizing stable ap and z at certain high spindle speeds.
5.
Conclusions
In this paper, a dynamics model and a predicting method for stability limits of high-speed finishing end milling were proposed using the mode synthesis technique based on the orthogonality of the multi-mode characteristics of MFOD system. The proposed method allows considering the effects of
The authors gratefully acknowledge the financial support of Natural Science Foundation of Shandong Province, PR China (Grant No. Y2007F41) and National Natural Science Foundation of China (Grant No. 50435020). Also, the authors wish to express gratitude to the engineer Mr. Z.W. Liu and the CNC machine center of SDU for their support.
references
Altintas, Y., Budak, E., 1995. Analysis prediction of stability lobes in milling. J. Annals CIRP 44 (1), 357–362. Budak, E., 2006. Analytical models for high performance milling, part II: process dynamic and stability. J. Int. J. Mach. Tools Manuf. 46, 1489–1499. Deshayes, L., 2007. Analysis of an equivalent tool face for the cutting speed range prediction of complex grooved tools. J. Mater. Process. Technol. 190 (1–3), 251–262. Gagnol, V., Bouzgarrou, B.C., Ray, P., Barra, C., 2007. Model-based chatter stability prediction for high-speed spindles. Int. J. Mach. Tools Manuf. 47, 1176–1186. Kelly, S.G., 1996. Mechanical Vibrations. McGraw-Hill, UK. Kivanc, E., Budak, E., 2004. Structural modeling of end mills for form error and stability analysis. Int. J. Mach. Tools Manuf. 44 (11), 1151–1161. Mane, I., Gagnol, V., Bouzgarrou, B.C., Ray, P., 2008. Stability-based spindle speed control during flexible workpiece high-speed milling. Int. J. Mach. Tools Manuf. 48, 184–194. Movahhedy, M.R., Mosaddegh, P., 2006. Prediction of chatter in high-speed milling including gyroscopic effects. Int. J. Mach. Tools Manuf. 46, 966–1001. Solis, E., Peres, C.R., Jimenez, J.E., Alique, J.R., Monje, J.C., 2004. A new analytical- experimental method for the identification of stability lobes in high-speed milling. Int. J. Mach. Tools Manuf. 44, 1591–1597. Tang, W.X., 2005. Research on High-speed Machining Stability and Dynamic Optimization. Ph.D. Dissertation. Shandong University, P.R. China. Weingaertner, W.L., Schroeter, R.B., Polli, M.L., Gomes, J.O., 2006. Evaluation of high-speed end-milling dynamic stability through audio signal measurements. J. Mater. Process. Technol. 1179 (1–3), 133–138. Yoon, M.C., Kim, Y.G., 2007. Chatter stability boundary analysis using RBNN. J. Mater. Process. Technol. 184 (1–3), 251–256.
journal homepage: www.elsevier.com/locate/jmatprotec
Prediction of chatter stability in high-speed finishing end milling considering multi-mode dynamics W.X. Tang a,∗ , Q.H. Song a , S.Q. Yu b , S.S. Sun a , B.B. Li a , B. Du a , X. Ai a a b
School of Mechanical Engineering, Shandong University, Jinan, Shandong 250061, PR China School of Software Engineering, Beijing University of Technology, Beijing, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history:
The strong demand for increasing productivity and workpiece quality in high-speed milling
Received 26 July 2007
make the machine-tool system has to operate close to the limit of its dynamic stability.
Received in revised form
This requires that the chatter stability is predicted accurately to determine the optimal
14 May 2008
milling parameters. An analytical stability prediction method was proposed with multi-
Accepted 7 June 2008
degree-of-freedom (MDOF) system modal analysis. This paper describes the development of this new method which allows considering the effects of multi-mode dynamics of system, higher excited frequency (i.e. tooth passing frequency) and wider spindle speed range on
Keywords:
stability limits in high-speed milling, and these to help in selection of milling parameters
Stability prediction
for a maximum material removal rates (MRR) in real operations without chatter. Some tests
High-speed milling
were carried out to demonstrate the quality of this method used in real machining. Final,
Chatter
the main influencing factors of stability limits in high-speed milling were analyzed. © 2008 Elsevier B.V. All rights reserved.
Multi-mode dynamics Frequency response function
1.
Introduction
High-speed milling is growing very rapidly in aerospace, automotive, die making and many other industries due to its advantages such as higher material removal rates (MRR), better surface finish and lower cost and so on. However, the unstable milling due to chatter vibration is not only one of the main limitations for productivity and workpiece quality, especially for a finishing high-speed milling, but also shorten the life of the machines and tools evidently. In particular, the aerospace, etc. industries have a strong demand for increasing MRR. Therefore, machine-tool system has to operate close to the limit of its dynamic stability. In addition, the manufactured structural parts in the aerospace industry become larger and more complex. If chatter vibration occurs during milling, the scrap rate of produced workpieces rises. This leads to immense economic losses. To run the milling process close
∗
to the limit between stable and unstable cut, the prediction of the stability limit is absolutely necessary. However, the predicting of stability in high-speed milling is more complicated than the conventional machining due to the rotating tool resulting in a time varying dynamics, periodical cutting forces and chip-load directions, and multi-degree-of-freedom (MDOF) structural dynamics. Thus, the investigation for predicting stability of high-speed milling against chatter vibration is vital for the goals of producing high quality products and maximum MRR in real machining. A wide range of research has been conduced to predict the stability with different methods, including analytical prediction method (Altintas and Budak, 1995; Budak, 2006), stability boundary analysis with the radial basis neural network (RBNN) (Yoon and Kim, 2007), experimental prediction method, such as using audio signal measurements, etc. (Weingaertner et al., 2006), and analyticalexperimental prediction method for stability lobes (Solis et al.,
Corresponding author. Tel.: +86 531 8839 5617; fax: +86 531 8839 6708. E-mail address: [email protected] (W.X. Tang). 0924-0136/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2008.06.003
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2004), etc. Deshayes (2007) analyzed an equivalent tool face for the cutting speed range prediction of complex grooved tools. Gagnol et al. (2007) elaborated a dynamic model of a highspeed spindle-bearing system on the basis of rotor dynamics predictions. Movahhedy and Mosaddegh (2006) developed the dynamics model of machine-spindle-tool-workpiece system considering the gyroscopic effects of spindle and used to predict the stability limits in high-speed milling. Mane et al. (2008) recently presented a stability-based spindle speed control method during flexible workpiece high-speed milling. These researches played an important part in practical manufactures. The stability of milling process is dependent on the dynamic behavior of machine-tool-workpiece system, which is often expressed in terms of the frequency response function (FRF) of system at the tool tip. The FRF of system at the tool tip is usually determined by considering the contribution of first or lower order mode characteristics of system only. At lower milling speeds, the effects of these higher order mode’s characteristics of system on the stability can be safely ignored. However, with the rising speed of the spindle, the increase of tool teeth pass frequency leads to the higher order modal frequencies of system are exited. The effects of multi-mode dynamics characteristics of MDOF system become more pronounced and the process stability deviate from the case expressed only by first or lower order modes of system. This may leads to a wrong prediction of the regions of a stable cutting process, especially at high-speed milling. It is therefore worthwhile to investigate as to how the stability during high-speed milling is crucially affected by the multi-mode characteristics of MDOF system and accurately predicted. In this paper, a prediction method of stability in highspeed finishing end milling is proposed with respects to the multi-mode dynamics characteristics of MDOF system using the mode synthesis technique. The proposed method was expected to provide practical guidance to machine-tool user for optimal process planning of spindle speeds and depth of cuts in milling operations. To verify stability lobes obtained by the proposed method, some milling tests were conducted. Several main influence factors of stability in high-speed milling were analyzed.
2.
Stability analysis for end milling
2.1.
Dynamics model and milling force
In order to compute the FRF at the tool tip, which is required for chatter stability analysis, a cross-sectional view at the contact zone of end tool and workpiece, which are included as part of multi-degree-of-freedom system, in high-speed milling is shown in Fig. 1 using the model presented by Altintas and Budak (1995). The analyzed tool with the diameter D and the number z of equally spaced teeth rotates at a constant angular velocity ˝ (rad/s). The radial immersion angle of the ith tooth varies with time as: ϕi (t) = ˝t + 2(i − 1)/z. The machine-toolworkpiece system is excited by the milling force F(t) at the tool and workpiece contact zone causing dynamic displacement
Fig. 1 – The cross-sectional view of an end mill in milling process.
q(t) of the structure governed by the following equation ¨ + C q(t) ˙ + Kq(t) = F(t) M q(t)
(1)
where M, C and K are the mass, damping, stiffness matrices of high-speed milling system respectively, which all are matrices of n × n orders (n = 1, 2, 3, . . .). For a point on the ith cutting tooth, the tangential (Ft,i ) and radial (Fr,i ) milling force components are assumed proportional to the chip-load defined by the chip thickness hi (t) and the axial depth of cut ap Ft,i (t) = Kt ap hi (t) Fr,i (t) = Kr Ft,i (t)
(2)
where Kt and Kr denote the special cutting force coefficients in the tangential and radial directions, respectively. And the tangential and radial forces can be resolved in the feed (x) and normal (y) directions Fx,i (t) = ıi (t)[−Ft,i (t) cos ϕi (t) − Fr,i (t) sin ϕi (t)] Fy,i (t) = ıi (t)[Ft,i (t) sin ϕi (t) − Fr,i (t) cos ϕi (t)]
(3)
where denoting a unit step function determining whether or not the ith tooth is cutting by ıi (t). The instantaneous chip thickness hi (t) consists of a static component due to the feed motion with feed per tooth ft , hs,i (t) = ft sin ϕi (t), and a dynamic component hd,i (t) which is generated due to the displacement of tool qtd,i (t) and the displacement of workpiece qwd,i (t). Because only the dynamic component hd,i (t) contributes to the dynamic chip-load regeneration mechanism, so hi (t) can be expressed as hi (t)
= hd,i (t) = qtd,i (t) − qwd,i (t) = ıi (t)[x(t, T) sin ϕi (t) + y(t, T) cos ϕi (t)]
(4)
where x(t, T) = [xt (t) − xw (t)] − [xt (t − T) − xw (t − T)] and y(t, T) = [yt (t) − yw (t)] − [yt (t − T) − yw (t − T)] describe the surface regeneration, i.e. the difference between the tool positions
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at the present and previous tooth passes; T = 2/(z˝) is the tooth passing period. And xt (t), xw (t), yt (t) and yw (t) present displacements of the tool and workpiece in x and y directions, respectively. Substituting hi (t) into Eqs. (2) and (3), the dynamic milling forces can be resolved in x and y directions and reduced as {F(t)} = 12 ap Kt [A0 ]{(t)}
(5)
where
z [A0 (t)] = 2
⎧ ⎪ A0xx = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪A = ⎪ ⎨ 0xy
1 2
A0xx A0yx
A0xy A0yy
cos 2ϕ − 2
(6)
Kr ϕ + sin 2ϕ Kt
{F(r) }eiωc t =
˛en ˛ex ˛en
1 Kr −2ϕ − sin 2ϕ + cos 2ϕ 2 Kt
˛ex
˛en
⎪ ⎪ A0yx ⎪ ⎪ ⎪ ˛ex ⎪ ⎪ ˛en ⎪ ⎪ ⎩ A0yy = 1 − cos 2ϕ − 2 Kr ϕ − Kr sin 2ϕ 1 Kr = 2ϕ − sin 2ϕ + cos 2ϕ 2 Kt Kt
2
Kt
(7)
det I −
(Kr − ωr Mr )˚r = 0 ˚Tr M˚r = I
(8)
Suppose the modes of system are not dense, and the eigenvalues can be arranged as 0 < ω1 < ω2 < · · · < ωr < ωr+1 < · · · < ωn
(9)
Based on the orthogonality of mode characteristics of MDOF system (Kelly, 1996), the dynamics equation of the rth order mode can be written as follows (r = 1, 2, · · ·, n)
= R + I = −
j=1
=0
(14)
n j=1,j = / r
[G(j) (iω)]
(11)
z ap Kt (1 − e−iωc T ) 4
(15)
2R (1 + ε2 ) zKt
(16)
sin ωc T I = R 1 − cos ωc T
(17)
where ε=
The above can be solved simply by finding the spindle rotating speed N and the tooth passing period T: N=
60 zT
T=
1 ( + 2k) ωc
(18) (k = 0, 1, . . .)
(19)
where = − 2cot−1 ε is the phase shift between the inner and outer modulations (present and previous teeth). Therefore, for given cutting geometry, cutting force coefficients, eigenvalue and chatter frequency, the corresponding stability limit axial depth of cut and spindle speed can be determine from Eqs. (16) and (18).
2.3. [G(j) (iω)] = [G(r) (iω)] +
where R and I are the real and imaginary parts of eigenvalue , respectively. And the final expression for chatter free axial depth of cut is given as
(10)
where Mr , Cr , Kr , Fr (t) and qr (t) are the modal mass, modal damping, modal stiffness matrices, modal forces vector and modal space vector (xr (t) = r qr (t)) of the rth order mode of system in the generalized co-ordinate, respectively. And the frequency response function of the n-DOF system can be expressed as
[G(iω)] =
1 (r) ap Kt (1 − e−iωc T ) G0 (iω) 2
and the eigenvalue of the characteristic equation is
ap = −
n
(13)
Eq. (12) has a non-trivial solution if its determinant is zero
˛ex
Chatter stability
Mr q¨ r (t) + Cr q˙ r (t) + Kr qr (t) = F r (t)
(12)
where {F(r) } represents the amplitude of the dynamic milling force {Fr (t)} at the tool- workpiece contact zone corresponding to the rth mode of the system. And assumed (r)
The characteristic equation of the multi-degree-of-freedom system expressed by Eq. (1) has n eigenvalues, on the other words, n natural frequencies (ω1 , ω2 , . . ., ωr , . . ., ωn ) and n eigenvectors ({1 }, {2 }, . . ., {r }, . . ., {n }). The rth eigenvalue ωr and eigenvector r satisfy the following eigen-equations and orthogonality conditions
1 ap Kt (1 − e−iωc T )[A0 ][G(r) (iω)]{F(r) }eiωc t 2
[G0 (iω)] = [A0 ][G(r) (iω)]
here, ˛en and ˛ex are entry angle and exit angle of the tool, respectively.
2.2.
where matrix [G(r) (iω)] which is the FRF of the rth order mode is defined between the tool and workpiece. The last item of above equation is the residue item except the rth order FRF and can be eliminated when the vibration occur at a dominate frequency ω which is around the rth order modal frequency ωnr of system based on above analysis. The vibrations are assumed to occur at a chatter frequency ωc , when a critical axial depth of cut is taken. The forces {Fr (t)} are described as a harmonic functions and the closedloop equation of the milling operation in the generalized co-ordinates and the Eq. (10) can be expressed in frequency domain as the following form (Tang, 2005)
Stability limits analysis
Based on the orthogonality of modal characteristics of MDOF system (Kelly, 1996), when vibration occurs at a chatter fre-
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quency ωc , which is around the rth order modal frequency ωnr , in high-speed milling process, the eigenvalue in Eq. (15) satisfies orthogonality condition Eqs. (8) and (9) and can be expressed as (Altintas and Budak, 1995): =
1
(20)
(r) [G0 (iω)] (r)
And [G0 (iω)] can be also expressed in the following form (r)
(r)
(r)
[G0 (iω)] = Re[G0 (iω)] + Im[G0 (iω)] (r)
(21)
(r)
where, Re[G0 (iω)]) and Im[G0 (iω)] are real and imaginary components of the FRF corresponding the rth order mode of the system. The FRF implicitly contains the mode characteristics, i.e. modal frequency (ωnr ), modal damping ratio ( r ) and modal stiffness (kr ), of the system. Once these characteristics are obtained analytically or experimentally, the both components can be determined as follows 1 − 2r
(r)
Re[G0 (iω)] = A0xx
2
−2i r r
(r)
Im[G0 (iω)] = A0xx
2
kr [(1 − 2r ) + (2 r r ) ]
2
2
kr [(1 − 2r ) + (2 r r ) ]
(22a)
(22b) Fig. 2 – Flow chart of obtaining stability lobes.
where r = ω/ωnr = f/fnr ; f = ω/2 is the excited frequency. By taking the average number za of teeth during the cut za =
zae 2D
(23)
where ae is the radial depth of cut. And the value of ae is small during a finishing end milling process and ae and ft both vary (r) within a narrow range. Then substituting za and [G0 (iω)] into Eqs. (20) and (16), the stable critical axial depth of cut is given by (r)
alim = −
1
(24)
(r)
2Kt za Re[G0 (iω)]
The equation shows that the stability limit of milling may (r) exist no other than the real components Re[G0 (iω)] of the FRF (r)
are less than zero, because of alim > 0 in practical milling processes. The differentiation about the parameter r of Eq. (22a) (r)
leads to the minimum of Re[G0 (iω)] when the condition r = 1/2 exists (1 + 2 r ) (r)
Re[G0 (iω)]min = −
1 [4kr r (1 + r )]
(25)
The minimum of the stability limits coordinating to all modes can be expressed as (r)
amin = min[amin ]
(r = 1, 2, · · ·, n)
(27)
The amin can be explained that the milling is stable at any spindle speeds as long as the axial depth of milling is less than the amin .
2.4.
Creating stability lobes diagram during milling
Based on the above analysis, the stability limits of high-speed milling processes can be predicted with respect to multiple mode characteristics of MDOF systems. To apply effectively the proposed method to predict the stability limits in real high-speed milling, the flow chart and process of creating stability lobes diagram using the proposed method are given and shown in Figs. 2 and 3, respectively: (1) Obtain the dynamic characteristics and milling processes parameters (fnr , mr , kr , r , Kt , Kr , ft , ae , z, za ) of the machinetool-workpiece system using experimental analysis; (2) Select one excite frequency (f) surrounding the rth order modal frequency (fnr ) of the system; (r)
So the minimum of the stability limits corresponding the rth order mode of the system can be written as
(r)
(3) Calculate the real (Re[G0 (iω)]) and imaginary (Im[G0 (iω)]) component of the frequency response function, Eqs. (22a) and (22b); (r)
(r)
amin =
4kr r (1 + r ) 2Kt za
(26)
(4) Calculate the stability limit (alim ) of milling, Eq. (26); (5) Calculate the spindle rotating speed (N) from Eq. (18) for each stability lobe k = 0, 1, 2, . . ., Eq. (19);
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Fig. 4 – Stability lobes obtained by predicting simulation.
Fig. 3 – Process of obtaining stability lobes.
(6) Repeat from Step (3) by scanning the excite frequencies; and, (7) Repeat from Step (2) by scanning the next order mode (fnr+1 ). To obtain the stability lobes of high-speed milling with respect to multi-mode characteristics of MDOF system, it is necessary to set an appropriate step load (f = fr+1 − fr ) and apply the above method repeatedly to all modes of FRF. And as a practical example, a stability lobes diagram, which based upon the modal parameters of a five-axis machining center (DMU-70V) with the solid carbide tool (diameter is 10 mm), is obtained by the proposed method and shown in Fig. 4. It should be noticed that the horizontal axis in Fig. 4 is the product of spindle speed (N) and number of tool teeth (z).
3.
Experimental validation
To validate the proposed stability predicting method, a series of experiments were performed on a five-axis machining center DMU-70V whose maximum spindle speed is 18,000 rpm. Three solid carbide end mills with one, two and four teeth (z = 1, 2, 4, D = 10 mm) were selected. The workpiece material was 0.45% C (24 HRC) steel. The modes characteristics of the machining center configuration are identified using the experimental modal analysis technique.
The test system is shown in Fig. 5. In Fig. 5, the symbols ux and uy denote the displacement responses measured by non-contact displacement sensors in feed (x) and normal (y) directions respectively. The ax and ay denote the acceleration responses measured by acceleration sensors. And fx , fy and fz denote the milling forces measured by a Kistler dynamometer in x, y and z directions respectively. In the milling tests, the selection of spindle rotating speeds (N) from 5000 to 17,000 rpm with the increments of 2,000 rpm was intended to verify the calculated stability limits for finishing end milling processes with different axial depths of cut (ap ). For each spindle speed, ap started with 2 mm and it was increased with the steps of 2 mm in each new cutting passing. Since the primary aim of the tests was to analyze and verify the stability of the finishing milling process where low immersions used usually in practice, the radial depth of cut (ae were selected as 0.3, 0.5 and 1 mm and the feed per tooth (ft ) was 0.1 mm/tooth in up- and down-milling. The effect of the ft on stability is small and can be ignored (Kivanc and Budak, 2004). The results of tests with three solid carbide tools (z = 1, 2 and 4) are shown in Figs. 6–8, and compared with the calculated results by the proposed method, respectively. The tag ‘’ represents a milling process without chatter and ‘×’ represents an unstable milling process with chatter. The curves represent the calculated critical stability limits. In Figs. 6–8, it can be observed that measured and calculated stability limits showed reasonable agreement and the configurations of stability lobes during milling process using three tools with different number (z) of tool teeth, z = 1, 2, 4, were clearly different even under same milling conditions, i.e. N, ap , ae and ft .
Fig. 5 – Testing system.
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4. Influencing factors of the stability in high-speed finishing end milling In order to obtain perfect surface finish, milling process has to be stable. The aim of predicting stability is to determine the optimum milling process parameters, i.e. N, ap , ae and ft , for a maximum material removal rate (MRR) without chatter. The MRR is proportional to the multiplication of milling parameters and the number of tool teeth z MRR = ap ae Nzft
Fig. 6 – Stability lobes for z = 1, ae = 0.3 mm, ft = 0.1 mm/tooth, up-milling.
(28)
In general, in a finishing end milling process, value of both ae and ft and their variable range are small. For simplicity, effects of both ae and ft on stability can be neglected. The detail of considering effects of both axial and radial depths of cut can be found in reference (Budak, 2006). Therefore, the rest of the analysis focuses on the mainly influencing factors except both ae and ft and their optimization to maximize the stability limits against chatter.
4.1.
Frequency response functions
The frequency response function implicitly contains the mode characteristics of the system, like the natural frequency (fnr ), modal stiffness (kr ) and modal damping ratio ( r ). If the mode characterisics of the system change, there is a direct consequence for the process stability. The optimum spindle speed ranges, in which a large depth of cut can be achieved without chatter, will also shift. If the modal stiffness or/and damping ratio are different, the minimum (amin ) of stabile axial depth of cut (ap ), which depended on the modal parameters closely, is also different. According to the Eqs. (22a) and (24), the larger are the modal stiffness (kr ) and modal damping ratio ( r ), the smaller is the (r)
Fig. 7 – Stability lobes for z = 2, ae = 0.3 mm, ft = 0.1 mm/tooth, up-milling.
absolute value of real component Re[G0 (iω)] of FRF, thus, the larger stability limit axial depth of cut (alim ) can be acquired. For example, the calculated and measured stability lobes both show that the ranges of spindle speed where the largest cutting depths can be achieved without chatter are different in Figs. 6–8, respectively (in Fig. 6: N = 11 krpm, alim = 14 mm; in Fig. 7: N = 16 krpm, alim = 14.5 mm; and in Fig. 8: N = 15 krpm, alim = 14.5 mm). The maximum MRR obtained during stable milling (N = 11 krpm, ap = 14 mm, z = 1) shown in Fig. 6 is only approximate 1/5 of the milling (N = 15 krpm, ap = 14 mm, z = 4) shown in Fig. 8. In practice, it is expected that machine-tool system operate at a range of spindle speed close to its permitted maximum speed without chatter for the aim of enhancing both the finish quality and MRR. Hence, optimizing the modal stiffness and damping ratio of higher order modes, which corresponding higher spindle speed zone, is more significant than other lower order modes in high-speed milling.
4.2. Fig. 8 – Stability lobes for z = 4, ae = 0.3 mm, ft = 0.1 mm/tooth, up-milling.
Modal frequency
In Fig. 3, it is observed that each stability lobe was located along abscissa (spindle speed N) in stability lobes diagram depended on the magnitude of corresponding a certain
j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 2585–2591
order mode’s natural frequency. For the same machine-toolworkpiece system, the stability lobes corresponding larger value of the modal frequencies are located at righter of the abscissa, i.e. higher spindle speed zone. This higher speed zone should be optimized emphatically for maximum MRR in real high-speed milling. Comparing between Figs. 6–8, it can be found that the maximum modal frequencies (fnr ) excited in available spindle speed range (0–18000 rpm) are respectively the first order fn1 in Fig. 6, the fourth order fn4 in Fig. 7 and the fifth order fn5 in Fig. 8 on a same machine-tool-workpiece system. The critical stable depth of cut (alim ) corresponding these different modal frequencies is also different. This may lead to an evident different MRR which can be achieved without chatter. Therefore, the actual multi-mode dynamics of the investigated machine-tool-workpiece system has to be considered in highspeed milling to obtain the improvement of the stability limits predicting accuracy and optimum milling process parameters.
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multi-mode dynamics of MFOD system, higher excited frequency (i.e. tooth passing frequency) and wider spindle speed range on stability limits in high-speed milling. The process and flow chart of predicting stability limits using the proposed method were presented for applying this method into the real milling operations. And the stability lobes diagrams were obtained using this method and validated by milling experiments. The use of the proposed method in prediction of stability contributes to obtain an improvement of the stability lobes accuracy and optimum milling conditions for enhancing the surface finish quality and MRR in practice. The results of calculated and experimented show that prediction of the stability limits, especially under higher spindle speed, required considering the actual multi-mode dynamics of the investigated machine-tool-workpieces system. Otherwise, unexpected chatter vibration may occur which will damage the workpiece, the tool, or the machine-tool.
Acknowledgements 4.3.
The number (z) of tool teeth
Since the tooth passing frequency is proportional to the number (z) of tool teeth, a tool with larger number (z) of teeth can be selected to increase the MRR. Simultaneous the higher order modal frequencies of system may also be excited in lower spindle speed milling. Although the difference of the number (z) of teeth hardly affects the dynamics of the whole system and the minimum of the stability limits, it may lead to the change of the location of every lobe at abscissa (N) in stability lobes diagram. Therefore, when the tools with different number (z) of teeth are used at the same milling condition, the configurations of stability lobes may appear obvious different. In order to assess the effects of different number (z) of teeth on the stability limits in high-speed milling, the stability lobes between three tools with different number (z) of teeth, z = 1, 2 and 4, are compared in Figs. 6–8. It can be found that the configurations of stability lobes, the maximum stable ap , N and MRR obtained in three cases using different tools (z = 1, 2, 4) were different evidently. A trend of increase of MRR with increasing number (z) of teeth at same milling parameters condition is existent, however, to obtain a maximum MRR without chatter, the optimum milling parameters must be determined using the stability lobes curve predicted accurately. Therefore, the importance of selecting number (z) of teeth is two folds. First, in some real cases, axial depth ap is fixed due to the geometry of the part, special for the finishing milling with low radial immersion, thus the maximum stable N and z must be determined. Second, the available maximum spindle speed N for a certain machine-tool is fixed, the maximum MRR can only be achieved by optimizing stable ap and z at certain high spindle speeds.
5.
Conclusions
In this paper, a dynamics model and a predicting method for stability limits of high-speed finishing end milling were proposed using the mode synthesis technique based on the orthogonality of the multi-mode characteristics of MFOD system. The proposed method allows considering the effects of
The authors gratefully acknowledge the financial support of Natural Science Foundation of Shandong Province, PR China (Grant No. Y2007F41) and National Natural Science Foundation of China (Grant No. 50435020). Also, the authors wish to express gratitude to the engineer Mr. Z.W. Liu and the CNC machine center of SDU for their support.
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