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Three dimensional chatter stability prediction based on the exponential cutting force model
Yiqing Yang, Quiang Liu, Songmei Yuan, Jokin Munoa
To cite this version:
Yiqing Yang, Quiang Liu, Songmei Yuan, Jokin Munoa. Three dimensional chatter stability prediction based on the exponential cutting force model. 1st International Conference on Virtual Machining Process Technology (VMPT 2012), May 2012, Montreal, Canada. �hal-01502409�
Three dimensional chatter stability prediction based on the exponential cutting force model.
Yiqing Yang1, Qiang Liu1, Songmei Yuan1, Jokin Munoa2
1 School of Mechanical Engineering and Automation, Beihang University, China 100191
2 Ideko-IK4 Technological Center, Elgoibar, Spain 20870.
yyiqing@buaa.edu.cn, qliusmea@buaa.edu.cn, yuansm@buaa.edu.cn,jmunoa@ideko.es
Abstract: Chatter stability prediction is widely used to avoid chatter which restricts the machining quality and productivity. A lot of works have been done to predict the stability chart fast and accurately. However, most of them are based on the linear force model, and the chatter stability limits is formulated as independent on the feedrate, which does not conform to the reality well. This paper intends to investigate the chatter stability prediction based on the exponential force model, which is linearized by the Taylor equation when calculating the directional coefficients. Meanwhile, the stability model is extended to the three dimensional, which is especially applicable for the ballend mills, bull-nose end mills and inserted cutter where chatter may be brought up in Z direction. Simulation results show that the exponential force model agrees with the measurements as well as the linear force model in the cutting force prediction, and it is able to demonstrate the effect of the feedrate on the stability limit.
Keywords: milling, chatter stability, three dimensional, exponential force model.
1. INTRODUCTION
Machining chatter causes poor product quality and reduces the industry productivity. A lot of works have been done to predict and avoid it, based on the chatter stability prediction. One of the most common methods is zero-order solution [Altintas, 2000]. It is fast and high accuracy except for some cases like highly interrupted milling [Altintas et al., 2008]. Later, G. Stepan improved the stability simulation accuracy of these cases by proposing the semi-discretization method [Insperger et al., 2002][Dombovari et al, 2012].
So far, most of the chatter stability prediction works are based on a linear cutting force model, where the cutting coefficients are treated as constants. One problem related with the linear cutting force model is that the stability lobes are independent on the feedrate. However, chatter occurrence is found to be related with the feedrate in reality
[Faassen et al., 2003]. Other available cutting force models have been investigated in the literature [Faassen et al., 2003] and [Budak, 2006], and exponential force model is the major one. Therefore, it is employed to calculate the stability lobes in this paper.
Munoa investigated the chatter stability prediction of face milling based on the exponential force model with an application to the single degree of freedom system [Munoa et al., 2006]. However, chatter may be brought up by the modes in the Z direction when using ball-end mills, bull-nose end mills and inserted cutters. Altintas has proposed a three dimensional model to calculate the stability based on the linear force model [Altintas, 2001], considering the X/Y/Z machine tool compliance. Campa applied a three-dimensional dynamic model to calculate the stability lobes of low rigidity parts with a corner radius tool [Campa et al., 2007]. Based on their work, the three dimensional stability is extended to the exponential force model.
This paper is organized as follows: basic chatter theory is introduced first, and the chatter stability prediction based on the exponential force model is formulated. The simulation results are compared with the linear force model and classic 2D chatter stability model. The results are concluded in the end.
2. BASIC THEORY OF THREE DIMENSIONAL CHATTER.
Regenerative chatter is the most common type of chatter, as a result of the previous and current vibration waves printed on the workpiece caused by the machine tool flexibility.
A detailed explanation can be found in the literature [Altintas, 2000].
Fig. 1: Dynamic model of the cutting tool.
Fig.1 illustrates a bull-nose end mill. The machine tool vibrates during the machining because of the flexibility of the structure, which cause the fluctuation of the chip thickness h. As the existence of the lead angle κ, h is affected by the vibration in Z direction. The chip thickness can be expressed as
sin( ) sin
( ))
( j c j j,0 j g j
h
c is the preset feedrate, j,0, j is the previous and current vibration waves of cutter. g(j) is a step function determining whether the tool is in or out cut.
The first term c sin(j) sinκ is a static component and does not contribute to the chatter, therefore it is neglected. The dynamic chip thickness can be written as
sin( ) cos( )sin cos
( ))
( j j j j
d x y z g
h (1)
where, △x=x(t)-x(t-T), △y=y(t)-y(t-T) , △z=z(t)-z(t-T). T is the tooth passing period. By projecting the cutting force into the Cartesian coordinate system, the X/Y/Z components are obtained as,
a t r
z y x
dF dF dF
dF dF dF
sin 0
cos
cos cos sin
cos sin
sin cos cos
sin sin
(2) The total cutting force can be calculated by summing the force components acting on each tooth.
3. CHATTER STABILITY PREDICTION BASED ON THE EXPONENTIAL FORCE MODEL
The cutting coefficients are expressed as an exponential function of the chip thickness h for the exponential force model. Such as,
r r ac
p r tc
q r rc
ae re te
h K K
h K K
h K K
K K
K 0
Then, the cutting force can be expressed as
r a
p q r t
a t r
h K
h h K a K F F F
1 1
1
Where, p, q, r are the index number of chip thickness in the tangential, radial and axial direction. The exponential function can be formulated in a variety of equivalent ways Particularly, it can be defined by the power series according to the Taylor equation. Let h=he+hd. he is the stationary solution, and hd is the dynamic component.
Then h1-p =(he+hd) 1-p=he 1-p+(1-p)he-phd+O(he) Neglecting the stationary component he 1-p
and higher order polynomials O(he) for the simplification of calculation, the cutting force acting on tooth j in the radial, tangential and axial direction is,
sin sin cos sin cos ( ) 1 )
1 ( ) 1 ( ) 1 (
j a r
r d p
d q
d T
a t r
g z y x
x K K
h r h
p h
q aK
F F F
(3)
Combining Eq.(2) – Eq.(3), it can be obtained as,
z y x A aK F
F F
T z
y x
(4) [A] is the directional coefficient factor matrix, which is a periodical function of tooth passing period. By using the Zero-order solution, the average term of [A] is kept, and Eq.(4) can be rewritten as
( ) ) 2( N aK t
t
F T
(5)
Therefore, the elements in [α] are all constants which are relied on the entry and exit angles. They are formulated as follows,
exstex st ex st ex
st ex st ex st ex st ex st ex st
q r
r a
E
q r
r a
E
q r
r a
E
p
q r
r a
E
p
q r
r a
E
p
q r
r a
E
p
q r
r a
E
p
q r
r a
E
p
q r
r a
E
c K
q c
K r
c K
q c
K r
c K
q c
K r
c p
c K
q c
K r
c p
c K
q c
K r
c p
c K
q c
K r
c p
c K
q c
K r
c p
c K
q c
K r
c p
c K
q c
K r
sin sin cos 1
sin sin sin 1
cos
sin sin cos 1
sin sin sin 1
cos sin
sin sin cos 1
sin sin sin 1
sin sin
sin sin sin 1
sin sin cos sin 1
sin sin cos cos 1
cos
sin sin sin 1
sin sin cos sin 1
sin sin cos cos 1
cos sin
sin sin sin 1
sin sin cos sin 1
sin sin cos cos 1
sin sin
sin sin cos 1
sin sin sin sin 1
sin sin sin cos 1
cos
sin sin cos 1
sin sin sin sin 1
sin sin sin cos 1
cos sin
sin sin cos 1
sin sin sin sin 1
sin sin sin cos 1
sin sin
33 32 31 23 22 21
13 12 11
(6)
After obtaining the directional factor, the formulation to calculate the eigenvalues and stability lobes are straightforward and can be referenced with bibliography [Altintas, 2001].
4. STABILITY LOBES.
4.1 Dynamic Chip Thickness:
Cutting tests are performed on a three axial milling machine by machining Aluminium to identify the cutting coefficients (Fig.2). The tool diameter is 12mm, and the teeth number is 2. The immersion angle κ = π/4. The cutting force is measured by Kistler 9257B. Cutting coefficients based on the linear and exponential cutting force models are both identified. See Table 1.
Fig. 2: Tests for the cutting coefficients identification
Table 1 Identified cutting coefficients for the linear and exponential force model Linear force model
Ktc 812.6 N/mm2 Kte 15.91 N/mm
Krc 0.4177 -- Kre 21.04 N/mm
Kac 0.1194 -- Kae 2.74 N/mm
Exponential force model
Kt 581.6 N/mm2-p p 0.2278 --
Kr 0.3323 mmq-p q 0.4637 --
Ka 0.0560 mmr-p r 0.4882 --
The cutting force under a series of cutting conditions are predicted and compared with the experiment measurements after identifying the cutting coefficients (Fig.3). It can be seen that both the force models fits the experiments well and are capable of modelling the milling process. Next, three dimensional chatter stability is predicted.
Fig. 3: Cutting force comparison: measurement (solid line), exponential force model (dashed), linear force model (dashdot). Slot milling. Tool diameter D=12mm, teeth
number N=2, κ= π/4. (a) n=4000rpm, F=800mm/min, ap=1mm; (b) n=4000rpm, F=1200mm/min, ap=1.5mm
4.2. Chatter stability simulation
Consider the case when there is one dominant mode in the X, Y and Z direction, and the modal parameters are m=0.03993kg, ξ=1.1%, ωn=922Hz. The tool parameters are identical with the one used in the cutting tests. Several milling scenarios are simulated.
Firstly, the two dimensional and three dimensional chatter stability models are compared based on the exponential force model. It can be seen that the three dimensional model reaches higher stability limit after introducing the mode in Z
Fig. 4: Comparison of two dimensional and three dimensional chatter stability based on the exponential force model. Half immersion, up milling. Tool diameter D=12mm, N=2,
κ= π/4.
direction (Fig.4). Specially when κ= π/2, the Z component in Eq.(1) is null, and the proposed three dimensional chatter stability model can be reduced to the two dimensional case.
Next, the feedrate effect on the stability is able to be demonstrated on the exponential force model. After implementing more simulations with different feedrate per tooth c in Eq.(6) (i.e. c=0.1, 0.2, 0.3), it is found that the minimum stability increases slightly as the feedrate increases. Besides, the three dimensional stability model based on the exponential force model is compared with the linear force model proposed by [Altintas, 2001]. In this case, the acquired stability lobes of linear force model are closer with the exponential force model using higher feedrate c (Fig.5).
Fig. 5: Chatter stability comparison between three dimensional exponential (different feedrate) and linear force model. Half immersion, up milling. Tool diameter D=12mm,
N=2, κ=π/4.
5. CONCLUSIONS
The chatter stability based on the exponential force model is proposed while the flexibility in X/Y/Z direction is considered. Simulation results show that the exponential force model can predict the cutting force as well as the linear force model, and it is able to predict the stability considering the effect of feedrate. Further experiments will be conducted to verify the three dimensional chatter stability based on the exponential force model. And the effect of neglecting the higher harmonics during the linearizing process of the exponential force model will be investigated.
6. ACKNOWLEDGEMENTS
This work is sponsored by the Significant Science and Technology Special Program of Machine tools (2010ZX04014-052, 2010ZX04014-017, 2011ZX04016-021) and the Fundamental Research Funds for the Central Universities (YWF-11-03-H-017).
7. REFERENCES
[Altintas, 2000] Altintas Y.; “Manufacturing Automation”, Cambridge University Press, Cambridge, 2000.
[Altintas, 2001] Altintas Y.; “Analytical Prediction of Three Dimensional Chatter Stability in Milling”, JSME-International Journal Series C: Mechanical Systems, Machine Elements & Manufacturing, 44/3, 2001, pp 717-723.
[Altintas et al., 2008] Altintas Y., Stepan G., Merdol D., Dombovari Z.; “Chatter stability of milling in frequency and discrete time domain”, CIRP Journal of Manufacturing Science & Technology, 1/1, 2008, pp 35-44.
[Budak, 2006] Budak E.; "Analytical models for high performance milling. Part I:
Cutting forces, structural deformations and tolerance integrity", International Journal of Machine Tools & Manufacture, 46/12, 2006, pp. 1478-1488.
[Campa et al., 2007] Campa F.J., López de Lacalle L.N., Lamikiz A.; "Selection of cutting conditions for a stable milling of flexible parts with bull-nose end mills", Journal of Materials Processing Technology,, 191/1-3, 2007, pp. 279-282.
[Dombovari et al., 2012] Dombovari Z., Munoa J., Stepan G.; "General Milling Stability Model for Cylindrical Tools", Procedia CIRP, 4, 2012, pp. 90-97.
[Faassen et al., 2003] Faassen R.P.H., Van der Wouw R.P.H., Oosterling J.A.J.;
"Prediction of regenerative chatter by modeling and analysis of high-speed milling", International Journal of Machine tools & Manufacture, 43, 2003, pp.
1437-1446.
[Insperger and Stépán, 2002] Insperger T., Stépán G.; "Semi-discretization method for delayed systems”, International Journal of Numerical Methods in Engineering, 55/5, 2002, pp. 503-518.
[Munoa et al., 2006] Munoa J., Zatarain M., Bediaga I., Peigne G.; “Stability study of the milling process using an exponential force model in frequency domain”, Proceedings of CIRP - 2nd International HPC Conference, 2006, Vancouver, Canada.
