Mechanistic Modeling of Milling Process Damping Including Velocity and Ploughing Effects

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ScienceDirect Procedia CIRP 56 (2016) 124 – 127

9th International Conference on Digital Enterprise Technology- DET2016 – “Intelligent Manufacturing in the Knowledge Economy Era

Mechanistic modeling of milling process damping including velocity and ploughing effects Min Wana*, Jia Fenga, Weihong Zhanga School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China

a

* Corresponding author. Tel./fax:+86-29-88495774.E-mail address:[email protected] (M. Wan).

Abstract In this study, a practical method for modeling of process damping in milling process is presented. The cutting forces are decomposed into two components, i.e., the shearing and ploughing forces. The tangential and radial shearing forces are modified considering the direction changing of cutting velocity. With the assumption of small amplitude vibration, ploughing forces are simplified as linear forms through introducing equivalent viscous damper. The effect of cutting velocity and the equivalent viscous damper are integrated into milling forces model since they all have contributions to process damping. The equation governing the dynamics of the milling system is constructed to predict the stability lobe. The prediction is verified by previous researchers’ experiments. It is shown that when the proposed process damping model is taken into consideration, the accuracy of stability prediction can be improved significantly at low cutting speeds. ©2016 2016The TheAuthors. Authors. Published by Elsevier B.V.is an open access article under the CC BY-NC-ND license © Published by Elsevier B.V. This Peer-review under responsibility of the Scientific Committee of the “9th International Conference on Digital Enterprise Technology - DET (http://creativecommons.org/licenses/by-nc-nd/4.0/). 2016. Peer-review under responsibility of the scientific committee of the 5th CIRP Global Web Conference Research and Innovation for Future Production Keywords: Velocity effect; Ploughing force; Process damping; Stability lobes

1. Introduction The occurrence of chatter vibration during milling process results in low precision of machining surface and even damages the machine spindle or the cutter. Therefore, selecting the chatter-free cutting parameters based on the predicted stability lobe is of great importance to achieve high material removal rate and high machining quality. The classical chatter theories can achieve good prediction of machining stability for relatively high speed cutting process. However, the critical depth of cut predicted by the classical metal cutting theories is much lower than the experimental values at low cutting speeds, since the additional damping generated in low speed cutting process will enhance the stability. Das and Tobias[1] expressed the traditional regenerative cutting force considering velocity effect. This modified cutting model only considers the change of direction in the velocity term and does not consider flank contact. Wu[2,3] assumed that the ploughing forces caused by this flank

contact are linearly proportional to the total volumes of the displaced material. He also calculated the total indented volume by establishing the relationship between the volume and the tool position. Chiou and Liang[4] simplified the indented volume as a linear model and analyzed chatter stability, it was shown that ploughing forces of worn cutters have positive damping effects that stabilize the cutting systems. Huang and Wang[5] extended the cutting force model including two cutting mechanisms and two process damping effects. Altintas et al.[6] identified the dynamic cutting force coefficients with a fast tool servo, and demonstrated that the process damping coefficient increases as the tool is worn. Recently, Ahmadi et al.[7,8] presented a new method in predicting the material dependent indentation coefficient using output-only modal analysis, and solved the stability of milling in frequency and discretetime domain. In this paper, the process damping is mainly consisted of the velocity and ploughing effect. The effect of cutting velocity and the equivalent viscous damper are integrated

2212-8271 © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the 5th CIRP Global Web Conference Research and Innovation for Future Production doi:10.1016/j.procir.2016.10.036

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Min Wan et al. / Procedia CIRP 56 (2016) 124 – 127

into milling forces model. Small amplitude vibration assumption is used to simplify the model. The stability of milling with process damping can be solved using updated semi-discretization method. For the machining process with damping effect, the predicted stability lobe is validated by the existing experiments from previous researchers’ works. Nomenclature Kt, Kr N i a Ft,s, Fr,s Ft,p, Fr,p hs, hd V t

߬

lw vc Ksp

ߤ 

st

tangential and radial cutting force coefficient tooth number index number of the tooth axial depth of cut theoretical tangential and radial shearing forces tangential and radial ploughing forces static and dynamic uncut chip thickness indented volumes cutting instant time tooth passing period tool wear land length cutting speed indentation constant friction coefficient extruded area under flank face feed rate of the cutter, mm/tooth

tangential direction of the cutter, is not same as the actual tangential shearing force. The slope of this cutting trajectory is

sin(Ti (t )) | tan(Ti (t )) ri (t ) / vc

where ri is the vibration velocity, which is coupled in two direction, and can be expressed as

ri (t )

x(t )sin(Ii (t ))  y(t ) cos(Ii (t ))

The total cutting forces in dynamic milling process are mainly consisted of two components, i.e., the shearing forces on the rake face and the ploughing forces on the flank face. The shearing force acts on the shear plane, and the ploughing force is a result of the extrusion and friction between the flank face of tool and machining surface.

(2)

The actual shearing forces which are determined by the cutting coefficients, the chip thickness, and the depth of cut, are resolved in the tangential and radial directions of the cutter, and the theoretical dynamic shearing forces can be written as

Fri ,s

[ K r ah cos(Ti (t ))  K t ah sin(Ti (t ))]g (Ii (t ))

Fti ,s

[ K t ah cos(Ti (t ))  K r ah sin(Ti (t ))]g (Ii (t ))

(3)

where Ԅi(t) is the angular position of the ith cutting edge. The window function determining whether the tooth is in or out of cut is expressed as

­1 Ist d Ii (t ) d Iex g (Ii (t )) ® ¯0 otherwise

2. Dynamic Milling Model Considering Process Damping

(1)

(4)

The uncut chip thickness is consisted of the static and dynamic part

h hsi  hdi

(5)

with Fti ,s Fcti ,s :

kx cx

vc  r

vc

y

Ti Ii

y

cy

Ii

Ti Fri ,s T i Fcri ,s r

x

ky

x

hsi

st sin(Ii (t )); hdi

ri (t )  ri (t  W )

where ri(t) and ri(t-ɒ) are the dynamic displacements of the cutter at the present and previous tooth periods, respectively. The magnitudes of the slope of the wavy surface and the dynamic chip thickness are smaller compared with cos(θi(t)) and the static chip thickness, respectively. By substituting Eq. (5) into Eq. (3) and neglecting the high order terms hdisin(θi(t)), the dynamic shearing forces can be simplified as follows:

Fri ,s | [ K r a(hsi  hdi )  K t ahsi sin(Ti (t ))]g (Ii (t )) Fig. 1. Process damping due to the velocity effect.

The schematic of a simple model of milling with the velocity effect is given in Fig. 1. The direction of the actual tangential shearing force Fcti ,s varies when the toolworkpiece structure vibrates during the cutting process. The actual tangential shearing force is along the direction of cutting velocity, while there is an angle Ʌi between the cutting velocity and tangential direction of the cutter. So the theoretical tangential shearing force, which is along the

(6)

Fti ,s | [ K t a(hsi  hdi )  K r ahsi sin(Ti (t ))]g (Ii (t ))

(7)

The above discussion is for the influence of velocity changing of shearing forces. When considering ploughing effect, we assume that the ploughing forces are proportional to the volume of material extruded under the flank face of the tool as following equations[2,3]:

Fri ,p

Ksp ˜V ; Fti,p

P Fri,p

(8)

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Min Wan et al. / Procedia CIRP 56 (2016) 124 – 127

with

V

p1xx

(9)

Sa

where S represents the extruded area. There can be multiple indentation regions along the flank face, depending on the cutting conditions, vibration frequency, and tool geometry[9]. These indentation regions are different when sharp tool or worn tool is used, or small vibration or large vibration is suffered. So the accurate computation of the indentation area is more complex. Chiou and Liang[4] expressed this indentation area using a linear form with small amplitude vibration assumption

lw2 r (t ) 2vc

S

(10)

The energy dissipated by the indentation force in one vibration cycle is equal with the energy dissipated by the viscous damper, Ahmadi and Ismail[10] used this equivalent viscous damper approach to estimate ploughing forces as

Fri ,p

Ksp alw2 4vc

( x(t )sin(Ii (t ))  y (t ) cos(Ii (t )))

(11)

where lw is the flank wear width, which is related to the edge hone on the tool, the clearance and separation angles. Resolving the cutting forces acting on each tooth in the x and y coordinates and summing them up, the total dynamic milling forces acting on the cutter can be expressed with the x and y components as

­ Fx ½ ® ¾ ¯ Fy ¿

ª  cos(Ii (t ))  sin(Ii (t )) º ­ Fti ,s  Fti ,p ½ (12) ® ¾  cos(Ii (t )) »¼ ¯ Fri ,s  Fri ,p ¿ 1 ¬ i

N

¦ « sin(I (t )) i

Due to continuous change in the angular position of the cutting edge, the chip thickness and the cutting direction is time-variant. The dynamics of milling process is expressed by the following coupled, delayed differential equations 2

X(t )  (C +

Ksp alw ast P1 (t )  P2 (t )) X(t )  KX(t ) vc 4vc

aH(t )( X(t  W)  X(t )) P(t ) P(t  W)ˈH(t ) H(t  W) with

P1 (t )

ª p1xx (t ) « p (t ) ¬ 1yx

p1xy (t ) º p1yy (t ) »¼

p1xy p1yx p1yy

N

1 mx

¦ g (I t sin(I (t )) sin(I (t ))( K sin(I (t ))  K

1 mx

¦ g (I t sin(I (t )) cos(I (t ))( K

1 my

¦ g (I t sin(I (t )) sin(I (t ))( K

1 my

¦ g (I t sin(I (t )) cos(I (t ))( K

i

t

i

i

N

i

i

i

t

N

i

i

i

cos(Ii (t ))  K r sin(Ii (t )))

t

i 1 N

i

i

i

t

cos(Ii (t ))  K r sin(Ii (t )))

i 1

p2xy (t ) º p2yy (t ) »¼

sin(Ii (t ))(sin(Ii (t ))  P cos(Ii (t )))

p2xy

cos(Ii (t ))(sin(Ii (t ))  P cos(Ii (t )))

p2yx

sin(Ii (t ))(cos(Ii (t ))  P sin(Ii (t )))

p2yy

cos(Ii (t ))(cos(Ii (t ))  P sin(Ii (t )))

H(t )

ª hxx «h ¬ yx

hxy hyx hyy

cos(Ii (t )))

sin(Ii (t ))  K r cos(Ii (t )))

p2xx

hxx

r

i 1

ª p2xx (t ) « p (t ) ¬ 2yx

P2 (t )

i

i 1

hxy º hyy »¼

N

1 mx

¦ g (I (t )) sin(I (t ))( K

t

cos(Ii (t ))  K r sin(Ii (t )))

1 mx

¦ g (I (t )) cos(I (t ))( K

cos(Ii (t ))  K r sin(Ii (t )))

1 my

¦ g (I (t )) sin(I (t ))( K sin(I (t ))  K

r

cos(Ii (t )))

1 my

¦ g (I (t )) cos(I (t ))( K sin(I (t ))  K

cos(Ii (t )))

i

i

i 1 N

i

i

t

i 1 N

i

i

t

i

i 1 N

i

i

t

i

r

i 1

The static component of the chip thickness in the former part of Eq. (7) is dropped from the expressions since it does not contribute to the dynamic chip load regeneration mechanism, while the latter part of Eq. (7) can not be dropped since it relates to vibration velocity. The stability of milling process can be solved according to Insperger and Stepan’s methods[11]. Eksioglu et al.[12] expressed dynamic milling force considering ploughing effect and extended the semi-discretization method in flexible milling systems. The same methods are used in this paper to predict the stability lobes.

(13) 3. Simulation and Experimental Results Ahmadi and Ismail’s[8] experiments are adopted to verify the predicted results. Half and full immersion upmilling were performed at constant feedrate of 0.05 mm/tooth. The milling tool was a 25.4 mm diameter endmill with a single carbide insert. The cutting force coefficients were identified as Kt=900MPa, and Kr=540MPa, respectively. The indentation constant and the friction coefficient for Aluminum workpiece were used as 1.5ൈ105N/mm3 and 0.3, respectively. The modal parameters are shown in Table 1.

Min Wan et al. / Procedia CIRP 56 (2016) 124 – 127 Table 1. Modal parameters used[8]. Modal Direction

Natural frequency (Hz)

Damping ratio (%)

Modal stiffness (N/Ɋm)

Structural damping (Ns/m)

Feed direction(X)

346

1.5

4.74

65.30

Normal direction(Y)

336

1.5

4.27

60.78

Fig. 2 shows the stability lobes computed using updated semi-discretization method for half immersion up-milling and full immersions. The red and black lines represent the stability lobes with and without process damping, respectively. The circles and crosses represent the experimentally observed stable and unstable points, which can be identified from the spectra of the measured cutting forces and sounds. When the effect of process damping is included, the accuracy of chatter predictions is improved significantly at low cutting speeds, and predicted result agrees well with experimental test.

D

mechanistic modeling of machining process damping with velocity and ploughing effects in end milling is presented. The comparisons between the predictions and experimental datas obtained by previous researchers’ experiments are in very good agreement, which verifies proposed model. It is shown that when the process damping effect is included in the chatter stability model, the accuracy of chatter stability predictions can be improved significantly at low cutting speeds. Therefore, the stability lobe obtained by the proposed method can be used as the guidance of milling operations in practice to avoid the trails. Acknowledgements This research has been supported by the National Natural Science Foundation of China under Grant No. 11272261 and the Program for New Century Excellent Talents in University under Grant No. NCET-12-0467 and the Fundamental Research Funds for the Central Universities. References

4 Analyt. w/P.D. Analyt. w/out P.D. Exp. Stable Exp. Chatter

3.5

Axial depth of cut[mm]

3 2.5 2 1.5 1 0.5

700

750

800

850 900 950 Spindle speed[rev/min]

1000

1050

1100

E 4 Analyt. w/P.D. Analyt. w/out P.D. Exp. Stable Exp. Chatter

3.5

Axial depth of cut[mm]

3 2.5 2 1.5 1 0.5

600

650

700

750

800 850 900 950 Spindle speed[rev/min]

1000

1050

1100

Fig. 2. Stability lobes with process damping for (a) half immersion upmilling, and (b) full immersion.

4. Conclusion Process damping is crucial for accurately predicting stability limits in low-speed machining. In this paper,

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