chang2009 Thrust force model for vibration-assisted drilling of aluminum 6061-T6

This model incorporates plowing force and strain rate-dependent shear strength to provide more accurate predictions than the existing model.. Wang et al.[13]analyzed the instantaneous un

Thrust force model for vibration-assisted drilling of aluminum 6061-T6

McMaster Manufacturing Research Institute (MMRI), McMaster University, 1280 Main St W., Hamilton, Ontario, Canada L8S 4L8

a r t i c l e i n f o

Article history:

Received 6 March 2009

Received in revised form

23 July 2009

Accepted 26 July 2009

Available online 5 August 2009

Keywords:

Drilling

Metal cutting

Vibration assistance

Ultrasonic assistance

Vibration-assisted drilling

Ultrasonic assisted drilling

a b s t r a c t

Vibration assistance has increasing applications in metal removal processes This method induces high-frequency and low-amplitude vibration in the feed direction during cutting, and has the potential to reduce cutting forces leading to improved surface quality and reduced tool wear Note that this cutting process is distinct from ultrasonic machining This paper presents a thrust force model to predict the thrust force during vibration-assisted drilling of aluminum 6061-T6 This model incorporates plowing force and strain rate-dependent shear strength to provide more accurate predictions than the existing model The results of 72 drilling experiments with TiN-coated standard twist drills are reported The predictions from the developed thrust force model are compared with the experimental results The comparison demonstrates that the maximum deviation between the predictions and the averaged values of the experimental measurements is 20% using the existing model and only 7% using the proposed model

&2009 Elsevier Ltd All rights reserved

1 Introduction

Conventional metal cutting methods, such as drilling, produce

relatively high cutting forces and low machined surface quality

High cutting forces generally increase tool wear, and reduce

machined surface quality This directly affects the post-processing

efforts such as surface finishing and deburring, leading to

increased production cost There are various methods to reduce

tool wear and improve surface finish These include using a

special coating on the tool; changing (typically reducing) the

material removal rate (MRR); or even laser-assisted machining,

which alters the mechanical properties of the workpiece material

One recent and promising technique is known as ultrasonic

assisted or vibration-assisted machining

Vibration-assisted machining is a pure mechanical process that

does not require sacrificing MRR or altering the mechanical

pro-perties of the workpiece material This technique typically induces

high-frequency (41000 Hz) and low-amplitude (o0.015 mm)

vibration in the feed direction of a cutting process It has been

shown that this technique can reduce thrust force and improve

surface quality One application of vibration-assisted machining is

vibration-assisted drilling (VAD)[1–5] We have previously shown

that under preferable vibration conditions, the thrust force can be

reduced by VAD, while poor choice of vibration conditions can

result in increase in thrust force [1] Modeling and predicting

thrust force is important for finding these preferable conditions

In general there are two methods to predict thrust force: finite element modeling and analytical modeling This paper presents the development of an analytical thrust force model for VAD that extends the existing model and provides more accurate predictions

Analytical thrust force models for conventional drilling are well established They include work by Wiriyacosol and Armarego [6], Armarego and Wright [7], Watson [8,9], Elhachimi et al [10,11], and Lopez de Lacalle et al [12] However, due to the dynamic nature of VAD, these models cannot be directly applied Wang et al.[13]analyzed the instantaneous uncut chip thickness

to model the thrust force and torque in VAD under different vibration frequencies Zhang et al.[14]used a similar approach to study VAD under different vibration amplitudes

Other related cutting force models include the dynamic cutting force model with the presence of regenerative vibrations or chatter vibrations Altintas[15] presented the dynamics of the general metal cutting process Budak and Altintas[16,17]studied and modeled the dynamic cutting forces for milling with chatter vibrations Their studies modeled the variation of uncut chip thickness by analyzing the instantaneous tool location and the profile of the machined surface Li and Li [18] modeled the dynamic cutting forces for milling by modeling the variation of uncut chip thickness and strain rate-dependent shear strength Roukema and Altintas [19–21] presented the modeling of the dynamic cutting forces for drilling by modeling the variation of uncut chip thickness Wu[22], Ismail et al [23], Chandiramani and Pothala[24], and Moufki et al.[25] modeled the dynamic cutting force with the presence of plowing The plowing forces are modeled analytically by determining the displaced volume of the

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International Journal of Machine Tools & Manufacture

0890-6955/$ - see front matter & 2009 Elsevier Ltd All rights reserved.



Corresponding author Tel.: +1 905 525 9140x27591; fax: +1 905 572 7944.

E-mail address: gary@mcmaster.ca (G.M Bone).

workpiece Lee and Altintas [26] and Wang and Zheung [27]

modeled the plowing force empirically, and verified that plowing

force can be significant

The prior model of VAD ignored both the plowing force

component and the strain rate dependence of the material This

paper presents a novel thrust force model for VAD that

incorporates these two factors Our modeling approach is

consistent with the approaches used in the modeling of dynamic

cutting force in metal cutting with the presence of chatter

vibrations discussed above It will be shown that by modeling

the dynamic uncut chip thickness alone as is done in the prior

VAD force model, the model fails to predict the thrust force for

VAD accurately, while the proposed model improved the accuracy

of the force predictions and the prediction of the favorable

vibration condition that minimizes thrust force In Section 2, the

theoretical development of the model is presented In Section 3,

comparisons between model predictions and experimental

mea-surements are presented Conclusions are given in Section 4

2 Thrust force model

2.1 Instantaneous axial uncut chip thickness

The effective (also known as dynamic) cutting geometries of a

drill vary with radius Therefore a drill is typically divided into

elements along the cutting lips direction and the elements are

analyzed individually using a conventional mechanistic cutting

model The total thrust force is obtained by summing all of the

thrust force components of the individual elements [6–12]

Following the methodology of the dynamic cutting force modeling

[13–27], the instantaneous uncut chip thickness for VAD can be

estimated by studying the instantaneous displacement and

velocity of the tool The displacement equals the summation of

the displacement due to the feed, Fnt, and the displacement due to

the vibration, A sin(2pft), as follows:

zðtÞ ¼ A sinð2pftÞ þ Fnt ð1Þ

where A and f are the vibration amplitude (mm) and frequency

(Hz), respectively; F is the feedrate (mm/rev); n is the spindle

speed (rev/sec); and t is the time (s) Similarly, the instantaneous

velocity is

_zðtÞ ¼ 2pfA cosð2pftÞ þ Fn ð2Þ

To determine the axial uncut chip thickness, which is critical to estimate cutting forces, it is necessary to monitor the maximum depth of removed material at the rotational location of interest,

zmax(y) Transforming the independent variable from time (t) to rotational angle of the drill (y) by substituting y¼2pnt into

Eq (1) gives

zðyÞ ¼A sin Fy

n

 

þfy

zmax(y) can be estimated by monitoring the axial location of the cutting lips prior to the current instant For a two-flute drill, where there are two cutting lips atpradians away from each other, all the axial locations of the cutting lips prior to current instant are equal to z(yqp), where q is a positive integer (seeFig 1) Therefore, the maximum axial location of the cutting lips prior to the current instant, i.e the depth of the materials immediately in front of the cutting lips at the current instant zmax(y) equals

zmaxðyÞ ¼zðympÞ ð4Þ

where m is the minimum positive integer that satisfies the expression

zðympÞ4zðy ½m þ 1pÞ ð5Þ

Nomenclature

hf, hl axial/dynamic uncut chip thickness (mm)

Tk axial location of segment k (mm)

zðtÞ; _zðtÞ axial displacement/velocity (mm, mm/s)

V cutting velocity (mm/s)

D diameter of the drill (mm)

DVi displaced volume (mm3)

b0 drill helix angle (rad)

2p drill point angle (rad)

W drill web thickness (mm)

lnd dynamic friction angle (rad)

gnd dynamic normal rake angle (rad)

fnd dynamic shear angle (rad)

Zd dynamic feed angle (rad)

ri effective radius of the ith element (mm)

F feedrate (mm/rev)

g flank clearance angle (rad)

Dli length of each segment (mm)

2W0 length of the chisel edge (mm)

mc mean coefficient of friction of the tool–work interface

zmax(y) maximum depth of removed material (mm)

M number of elements on each cutting lip

Ddk plowing depth (mm)

Fpt, Fpx plowing force in thrust and horizontal direction (N)

Fl,C, Fl,T principle cutting force (N)

DPi resultant thrust force for the ith element (N)

y rotational angle of the drill (rad)

Dy rotational difference between each segment (rad)

ti shear strength (MPa)

fsp specific plowing force (N/mm3)

n spindle speed (rev/s)

e; _e strain, strain rate (s1)

t time (s)

k total number of segment on each element

Dw width of each segment (mm)

A vibration amplitude (mm)

f vibration frequency (Hz)

For the example shown inFig 2, zmax(y) ¼ z(yp) and m ¼ 1.

The axial uncut chip thickness hftherefore equals

hf¼ zðyÞ zmaxðyÞ if zðyÞ4zmaxðyÞ

0 otherwise



ð6Þ

The dynamic uncut chip thickness hlcan then be calculated by

analyzing the drill geometry as in[6–9]:

hl¼hfsinðpÞ cosðz=2Þ ð7Þ

where

z¼tan1½tanðoÞcosðpÞ ð8Þ

o¼sin1 W

ri

 

ð9Þ

In Eqs (7)–(9), p is the drill point angle, W is the drill

web thickness, and riis the effective radius of the ith element

(seeFig 3)

2.2 Plowing force model

Because of the oscillation of the drill during VAD, a wavy

machined surface is produced after each half revolution of the

drill When the cutting edges engage the workpiece again,

plowing can occur, as shown in Fig 4 If the volume of the

workpiece displaced by the tool is known, the plowing force can

be estimated Based on the analysis by Wu [22], the resulting forces for the ith element are

In Eqs (10) and (11), Fptand Fpxare the plowing force components

in the thrust and horizontal directions, respectively; fsp is the experimentally determined specific plowing force; DVi is the displaced volume; andmcis the mean friction coefficient of the tool–work interface.DViwill now be estimated by considering the tool profile and the maximum depth of the machined surface The tool profile can be modeled by dividing each drill element into small segments along the drill flank direction.Fig 5shows the difference between the drill elements and the segments The axial location of segment k on each drill element can be calculated using

Tk¼zðyÞ  kDlitanðgÞ

sinðpÞ cosðz=2Þ ð12Þ

where k refers to the kth segment, Dli is the length of each segment, andgis the flank clearance angle The geometry of a drill element is shown is Fig 6 At any instant, the depth of the machined surface corresponding to each drill element is given by

zmax;k0¼A sin f ðyjpkDyÞ

n

þFðyjpkDyÞ

2p

k ¼ 1; 2; ; k ð13Þ

In Eq (13),Dyis the rotational difference between each segment (seeFig 5), k is the total number of segments on each element, and j is the minimum positive integer that satisfies the

Fig 4 (a) After the first cut (half revolution for a two-flute drill), a wavy machined surface is formed (b) In subsequent cut, the flank surface of the drill may come in contact with the machined surface (even at multiple locations), causing plowing of material.

Fig 2 Schematic of determining z max (y) of the tool.

A sin f ðyjpkDyÞ

n

þFðyjpkDyÞ

2p 4A sin

f ðy ðj þ 1ÞpkDyÞ

n

þFðy ðj þ 1ÞpkDyÞ

The theory behind Eq (14) is same as the theory behind Eq (5)

The plowing depth of the kth segment on a drill element can then

be calculated:

Ddk¼ Tkz0

max; k if Tk4z0

max;k

0 otherwise



ð15Þ

The displaced volume per segment can then be calculated by

multiplyingDdkwithDliand the width of the element,Dw:

DVi¼Xk

where

Dli¼W þ ½W coso0þ ðD=2Þcoso0 ði  1=2ÞDwtanO

Dw ¼ D coso0D0coso0

o0¼tan1 2W

D

 

ð19Þ

and

o0

¼sin1 W

W0

 

ð20Þ

where M equals the number of elements in one cutting lip, D is the diameter of the drill, and W0equals half the length of the chisel edge (seeFig 3)

2.3 Strain rate-dependent shear strength model

During metal cutting, the strain rate along the primary and secondary shear zone reaches 103–106s1 These strain rates are significantly higher than the nominal strain rate (103–101s1) used to determine generic material properties Therefore, instead

of using the generic value, the Johnson–Cook model will be used

to estimate the shear strengthti The empirical constants used in the Johnson–Cook model for aluminum 6061-T6 were reported by Guo[28] These constants were fine-tuned slightly using some of the experimental results, which will be presented in Section 3 In the proposed model,tiis modeled by the following equations:

ti¼1

2ð247:5 þ 77:4e0:676Þ 1 þ C ln e_

0:01

 

ð21Þ

C ¼ 0:058  0:194hl0:003V ð22Þ

e¼ cosgnd

sinfndcosðfndgndÞ ð23Þ

_

e¼ V cosgnd

0:005 cosðfndgndÞ ð24Þ

In Eqs (21)–(24),tiis the ultimate shear strength (MPa),eand _e

are the strain and strain rate, respectively, V is the instantaneous cutting velocity (mm/s), and C is a unitless parameter At a higher vibration frequency, because of the increase in strain rate, the shear strength of the material will be increased, resulting in a higher thrust force

Fig 6 Geometry of one element on the drill cutting edge.

Fig 5 Geometry of drill elements and segments.

2.4 Thrust force model

The thrust force for VAD can be predicted by combining the

results of Sections 2.1–2.3 with a conventional mechanistic model

The principle cutting forces along the cutting lips can be

calculated as discussed in[29]:

Fl;C¼ thlDw cosðlndgndÞ

sinfndcosðfndþlndgndÞ ð25Þ

Fl;T¼ thlDw sinðlndgndÞ

sinfndcosðfndþlndgndÞ ð26Þ

In Eqs (25) and (26),lndis the dynamic friction angle,gndis the

dynamic normal rake angle, andfndis the dynamic shear angle

Based on the methodology from [6–9], these angles can be

calculated using the following equations:

kr¼tan1½tanðpÞ cosðoÞ ð27Þ

ri¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

D coso0 i 1

2

 

Dw

þW2

s

ð28Þ

b¼tan1 2ritanðb0Þ

D

ð29Þ

gf ¼tan1 tanðbÞsinðoÞ

sinðpÞ  cosðpÞ sinðoÞtanðbÞ

ð30Þ

Zd¼tan1 _zðtÞ

2prin

 

ð31Þ

krd¼tan1 sinðkrÞ

cosðkrÞcosðZdÞ þtanðlsÞsinðZdÞ

ð32Þ

lsd¼sin1½cosðpÞ sinðZdÞ þsinðpÞ cosðoÞcosðZdÞ ð33Þ

gnd¼tan1 tanðgfdÞcosðlsdÞ

sinðkrdÞ þ

sinðlsdÞ tanðkrdÞ

ð35Þ

lnd¼p

6þ

gnd

fnd¼p

In Eq (29),b0is the drill helix angle Finally, the resultant thrust

force for each element can be calculated as

DPi¼Fl;Tsin p cosZdþFl;CsinZdþFpt ð38Þ

The total thrust force can be found by adding all the elemental

thrust force components together

3 Experimental apparatus

3.1 Vibration-assisted workpiece holder

In order to perform experimental studies on VAD, it is

necessary to design an apparatus to produce the necessary axial

vibrations between the tool and the workpiece that is compatible

with conventional CNC machining center In vibration-assisted

turning, vibrations are typically generated on the tool Designing a

vibration-assisted tool holder for drilling is not trivial because of

the connection needed between the rotating spindle and the

external power source of the vibration actuator Moreover,

because of the length of the tool, vibrating the tool axially also potentially increases tool wobbling In this study, it is more desirable to produce axial vibrations on the workpiece While the relative displacement between the tool and the workpiece remains unchanged, this approach provides a simple design solution and reduces the potential of tool wobbling A custom-designed piezoelectric actuated vibration-assisted workpiece holder was designed and tested for this purpose.Fig 7 shows the schematic and a photograph of the workpiece holder Details

of the mechanical and electrical design of this hardware are presented in Chapter 5 of[30]

3.2 Experimental setup

Experiments have been conducted to verify the accuracy of the developed model Each test was performed on a Makino MC56-5XA horizontal CNC machine tool, with the vibration-assisted workpiece fixture attached onto a table dynamometer, which was attached onto the machine table The vibration amplitude was chosen to be 0.002 mm, while the vibration frequency was varied from 4 to 12 kHz in 2 kHz increments Note that the static stiffness

of a typical 5-axis horizontal machine center is 30 kN/mm[31], and the weight of the machine table is 58 kg At the magnitude of the induced vibration force, the resultant vibration amplitude of the machine table at the lowest testing frequency (4 kHz) is 2.76  106mm, and therefore is negligible The drills used were 4.0 mm TiN-coated standard twist drills The cutting conditions chosen were 4000 rpm spindle speed and 0.06 mm/rev drill feed The workpieces were 3.18-mm-thick Al 6061-T6 plates (25 mm  25 mm) Four drilling experiments were performed for each cutting and vibration conditions Thrust forces were measured using Kistler Type 9255B table dynamometer with sampling frequency equal to 10 times the corresponding vibration frequency to avoid aliasing The dynamometer has a natural frequency of 2 kHz and stiffness of 3000 kN/mm (Kistler Instru-mente AG[32]) At the magnitude of the induced vibration force, the resultant vibration amplitude of the dynamometer at the lowest testing frequency (4 kHz) is 1.22  105mm, and again

is negligible The mean values of the thrust forces for each cutting and vibration condition were calculated for subsequent comparison with the mean values of the corresponding model predictions

4 VAD results and discussion

The experimental mean thrust force results are compared with the corresponding simulation results in Fig 8 The solid data points represent simulation results (based on Section 2), and the hollow circular data points represent experimental data The maximum deviation between experimental results and the simulations was 10% When the vibration frequency equals

10 kHz, the thrust force is minimized, with a reduction of 16%

Fig 7 Vibration-assisted workpiece holder.

Note that the thrust force reduction is dependent on the material

and cutting conditions

Fig 9presents a comparison between the mean value of the

experimental thrust force measurements (averaged over each set

of four tests) and simulation results with both the plowing force

and strain rate-dependent shear strength models (simulation),

without the plowing force model (w/o Fpt), and without plowing

force and shear strength models (w/o Fpt andti) Note that the

model w/o Fptandtirepresents the simulation results of the prior

models The comparison shows that the mean thrust force

predictions fall within 20% of the experimental measurements

when using the prior models, and 7% using the proposed model

The plot also shows that at higher vibration frequencies including

tibecomes more important This is logical since increasing the

vibration frequency increases the strain rate Both experimental

and simulation results consistently showed that a favorable

vibration frequency (10 kHz in this case) that minimize thrust

force exists Note that we have obtained similar results for other

drills and cutting conditions

5 Conclusions

A novel analytical thrust force model for VAD has been

proposed Instead of modeling only the variation of dynamic

uncut chip thickness, the proposed model incorporates the

presence of plowing forces and the variation of shear strength

due to the changes in strain rate in VAD with the variation of dynamic uncut chip thickness

Comparisons between experimental results and model predic-tions have shown the reliability of the developed model The mean values of the experimental thrust forces for all tested conditions fall within 77% of the model predictions, and the worst case error is 10% The comparisons also demonstrated the importance of modeling the plowing force and incorporating strain rate-dependent shear strength for producing accurate thrust force predictions The proposed model predicted the vibration frequency where thrust force is minimized, within the frequency resolution employed in our experiments (2 kHz) This work is important for industrial applications because reducing thrust force can reduce tool wear, reduce burr size, and increase the quality of the machined surfaces Moreover, because producing the vibration may be difficult, the proposed model can also assist the user to determine if the effort required to achieve the desired thrust force reduction is reasonable

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