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International Journal of Mechanical Sciences 63 (2012) 26–36
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International Journal of Mechanical Sciences
0020-74
http://d
n Corr
E-m
journal homepage: www.elsevier.com/locate/ijmecsci
Analytical modeling and experimental investigation of ultrasonic-vibrationassisted oblique turning, part III: Experimental investigation
H. Razavi a, M.J. Nategh a,n, A. Abdullah b
a Tarbiat Modares University, Mechanical Engineering Department, Tehran, Iranb Amirkabir University of Technology, Mechanical Engineering Department, Tehran, Iran
a r t i c l e i n f o
Article history:
Received 3 January 2012
Received in revised form
27 May 2012
Accepted 10 June 2012Available online 20 June 2012
Keywords:
Ultrasonic vibration
Vibration cutting
Oblique turning
Cutting forces
03/$ - see front matter & 2012 Elsevier Ltd. A
x.doi.org/10.1016/j.ijmecsci.2012.06.007
esponding author. Tel.: þ98 21 82884396.
ail address: [email protected] (M.J. Nateg
a b s t r a c t
The experimental investigation of the kinematics and dynamics of ultrasonic-vibration assisted oblique
turning (oblique UAT) has been undertaken in the present study. For the purpose of comparison, the
conventional oblique turning experiments were also carried out in parallel to the oblique UAT
experiments. The burnishing effect of the vibrating motion of the cutting tool has been studied as a
means of illustrating the kinematic interaction between the cutting tool and the workpiece in vibration
cutting. The experimental investigation of the influence of oblique UAT parameters on the dynamics of
the process has been another subject of the present study. The oblique UAT parameters taken into
consideration consisted of vibration amplitude, cutting speed, inclination angle, tool cutting edge angle
and feed rate. The cutting force components were differently influenced by these parameters. The
inclination angle, tool cutting edge angle and feed rate did not have any significant effect on the cutting
force ratio whereas the cutting speed and the vibration amplitude highly influenced this ratio. The
experiments were carried out with aluminium 6061 and 2024.
& 2012 Elsevier Ltd. All rights reserved.
1. Introduction
In an earlier study [1], the authors developed a kinematicsmodel for the ultrasonic-vibration assisted oblique turning (obli-que UAT) based on the assumption that the cutting tool neverdisengage from the workpiece during the process. Instead, thecutting tool presses against the lateral surface when it withdrawsfrom the cutting zone at the beginning of the noncutting part ofthe vibration cycle or when it approaches this zone for restartingthe cutting operation. This repeats in each cycle of vibrationwhich is applied to the cutting tool along the cutting velocity,leaving a toothed pattern on the lateral surface. The lateralsurface should locally harden due to the pressing motion of thecutting tool. The hardening effect is expected to constantly takeplace at different levels of vibration cutting parameters. Thehardening effect of the lateral surface and the toothed patternleft on this surface by the cutting tool motion have been verifiedin the present study for different values of vibration cuttingparameters including cutting speed, feedrate, and vibrationamplitude. It should be noted that the lateral surface is at angleof ð180�KrÞ
3 to the positive z-axis of lathe when feed direction istowards the spindle; Kr is the tool cutting edge angle. The lateralsurface is continuously generated and removed during the
ll rights reserved.
h).
operation. The pressing motion of the cutting tool and conse-quently the hardening effect is, of course, specific to UAT and doesnot happen in conventional turning (CT).
An additional objective of this study is to experimentallyinvestigate the oblique cutting forces generated in vibrationturning and also to verify the cutting forces estimated by thedynamic model developed by the authors [2] for oblique UAT.
In parallel to vibration cutting experiments, conventionalturning (CT) experiments were also carried out and the resultshave been compared with UAT.
2. Experiments
The experimental tests carried out in this study are describedin two sections. The hardness tests for the lateral surface and thetoothed pattern being left from the cutting tool motion aredescribed as the kinematics experiments in the first section. Thetests carried out for the cutting forces are presented in the nextsection.
2.1. Kinematics experiments
The surface hardness of the lateral surface is measured in bothCT and UAT in order to find the effect of tool vibration on thesurface hardness in different conditions. In order to compare thesurface hardness of the lateral surface in UAT and CT, it has been
Nomenclature
a Vibration amplitude along the cutting velocity (mm)A, B,C, m, n material parameters (in Johnson–Cook constitu-
tive model)b Depth of cut (mm)d Workpiece diameter (mm)f Vibration frequency (Hz)i Inclination angle ðdegÞKr Tool cutting edge angle ðdegÞN Spindle rotational speed ðrev=minÞR T XðtÞ Cutting force applied on tool in radial direction (N)
R T YðtÞ Cutting force applied on tool in tangential direction
(N)R T ZðtÞ Cutting force applied on tool in negative feed direc-
tion (N)T Work material temperature ð3CÞTmelt Melting temperature ð3CÞTroom Room temperature ð3CÞVC Cutting speed ðm=sÞ
Vf Feed rate ðm=sÞ or (mm/rev)xyz Machine tool coordinate system: x radial (or depth of
cut) direction, y tangential (or cutting velocity) direc-tion and z axial (or negative feed) direction
x0 0
y0 0
z0 0
Oblique coordinate system (rotated x0
y0
z0
system byangle i); x
00
along the tool cutting edge, y0 0
placed onthe lateral surface and z
00
normal to the lateral surfaceX00W=T ðtÞ Workpiece’s position relative to tool in cutting tool
edge directionY 00 W=T ðtÞWorkpiece’s position relative to tool in lateral surface
and perpendicular to tool cutting edge (mm)Z00 W=T ðtÞWorkpiece’s position relative to tool perpendicular to
tool cutting edge and normal to lateral surface (mm)bmean Average friction force angle between tool and chip
ðdegÞgn Normal rake angle ðdegÞe Equivalent plastic strain_e Plastic strain rate (s�1)_e 0 Reference strain rate (s�1)l Cutting force ratio defined as the ratio of resultant
cutting forces created in UAT and CT (RTUATRTCT
)mmean Average coefficient of friction between chip and tool
rake faces Huber-von misses equivalent flow stress (MPa)jn Normal shear angle (measured in normal plane) ðdegÞ
Fig. 1. Setup for kinematics experiments.
H. Razavi et al. / International Journal of Mechanical Sciences 63 (2012) 26–36 27
tried to achieve similar test conditions. For this purpose, a 6061aluminum rod of 70 mm was roughly turned into a rod of 68.6 mmin diameter to eliminate the rod run-out. The cutting parameterswere N¼ 160 rev=min, Vf¼0.2 mm/rev and b¼0.7 mm. The rodwas then finished into a rod of 68 mm in diameter (with cuttingparameters of N¼ 80 rev=min, Vf¼0.05 mm/rev, b¼0.3 mm) toeliminate the adverse effects of the previous roughing operationsuch as the uneven surface roughness and hardened surfaces due tothe high cutting forces involved in roughing operations and residualstress effects which cause some deviations from raw materialproperties. It is noteworthy that in some cases built up edge wascreated when machining aluminum 6061, but did not influence thekinematics of cutting.
A constant cutting depth of b¼2 mm was used in the experi-ments. For this purpose, several slots with 5 mm width and 2 mmdepth were cut into the rod (N¼ 80 rev=min, Vf¼0.05 mm/rev).The slots were 10 mm apart from each other. The followingmachining conditions were applied for both UAT and CT: rotationalspeeds at three levels (VC¼0.18, 0.36 and 1.2 m/s) and feed rates atfour levels (Vf¼0.05, 0.11, 0.25 and 0.4 mm/rev). The vibrationcutting was carried out with f¼20 kHz, a¼10 mm. In the main UATand CT tests, the aluminum rod was turned from a diameter of 68 to64 mm. The cutting condition was orthogonal ðKr ¼ 901 , i¼ 01Þ.The experimental setup is shown in Fig. 1. A horn designed andmanufactured in a previous study [3] was used together with atitanium-nitride carbide insert (VBMT�160404). The ultrasonicvibration was applied in the direction of the cutting velocity.
The 2 mm depth of cut chosen for machining experiments issufficiently large in comparison with tool nose radius of r¼0.4 mmand the measurement of surface hardness has been done inmachined regions cut by the straight part of the cutting tool edge.
In order to investigate the simultaneous effects of the cuttingvelocity and feed rate on the surface hardness of the lateralsurface in UAT, both the cutting and feed motions should stopsimultaneously. The CNC TM40 turning machine has been usedfor the experiments. In CNC machines the cutting and feedmotions are controlled independently by their individual drivers.Therefore, the cutting motion may continue some while when
feed motion stops, and vice versa. It should be noted that therevolution of spindle after feed stop leads to the elimination ofthe effects desired for the present experiments on the lateralsurface. The experimental tests indicated that in low rotationalspeeds or high machining forces, the cutting and feed motionsstop rather simultaneously. However, in smaller machining forcesor higher spindle speeds, it was observed that after feed stop andturning the machine off, the spindle continued its rotation up toabout 1/4 to 1/3 of one complete revolution. Care was taken toexclude those parts of the lateral surface machined during thisextra revolution from the experimental tests.
Each experiment was repeated three times and machinedsamples containing the lateral surface were cut off and preparedfor hardness measuring tests. The Bohler Vickers micro-hardnessdevice was employed to measure surface hardness. The device wasfirst calibrated for the hardness test range. The loads of 50, 100 and200 g were applied each for a time interval of 15 s. A fixture wasmanufactured for clamping samples under microscope. The hard-ness measuring setup is shown in Fig. 2. The tests were done fordifferent regions of the lateral surface of each sample several timesand the average Vickers hardness was obtained for each sample.
Table 1The dimensions and density of workpiece.
Inner diameter f192 mm
Outer diameter f194, 196 mm
Tube thickness 1, 2 mm
Density 2770 kg/m3
Fig. 3. Experimental setup in oblique UAT.
Fig. 2. Measuring hardness of lateral surface.
Table 2Oblique UAT parameters
Factors Levels
Inclination angle i 0,15,30,45 deg
H. Razavi et al. / International Journal of Mechanical Sciences 63 (2012) 26–3628
The Olympus optical microscope also has been used to study thesurface texture.
Fig. 4. Tooling systems used in oblique UAT experiments.
Tool cutting edge angle
Kr
60,75,90 deg
Cutting speed VC 0:53,0:8,1:3,2:1 m=s
Feed dare Vf 0:05,0:4,1 mm=rev
Vibration amplitude a 6, 11, 16 mm
Depth of cut b 1, 2 mm
Fig. 5. Eddy current setup for measuring the vibration amplitude of cutting
tool tip.
2.2. Oblique cutting forces experiments
The objective of these experiments were to obtain the cuttingforces being created in oblique CT and UAT processes with singlestraight edge cutting tool. In order to eliminate the tool noseradius effect, a thin tube of Al 2024 was used for the experiments.It should be noted that no built up edge (BUE) was observedunder the cutting parameters selected for machining Al 2024material. The dimensions of the workpiece and the density of Al2024 are given in Table 1.
The thickness of the workpiece was considered to be smallenough to ignore the variation of the cutting velocity along toolcutting edge. The inner and outer surfaces of the tube had to bemachined precisely for such a small thickness. In order to increasethe resistance of the thin tube workpiece against the machiningforces, a solid rod of Teflon material was inserted into the tube bya press fit. It should be mentioned that, in order to cut only thealuminum tube during the experiments, the Teflon rod was pre-machined through slot cutting operation along its length onregularly intervals to yield unsupported regions when insertedinto the tube. The machining experiments were carried out on thetube only along these unsupported regions. The relevant experi-mental setup is shown in Fig. 3.
To carry out the experimental tests, TM40 CNC lathe, Kistlerdynamometer 9257B, a power supply and an ultrasonic transducerhave been used. The tool geometry and cutting parameters usuallyemployed in the turning operations were selected for the obliqueUAT experiments as presented in Table 2. These parameters havebeen categorized in different levels for the purpose of designing theexperiments.
Several horns were designed and manufactured to work at afrequency of about 20 kHz and be able to accommodate cuttinginserts with different inclination and cutting edge angles asillustrated in Fig. 4.
A calibrated eddy current device was used according to Fig. 5 fornon-contact measurement of the tool vibration amplitude. It should
H. Razavi et al. / International Journal of Mechanical Sciences 63 (2012) 26–36 29
be mentioned that the actual value of the vibration amplitude ismeasured and then employed in the theoretical analysis.
Using the design of experimen algorithm, D-optimal responsesurface method (RSM-D Optimal), eighty eight tests weredesigned to cover the whole range of the working condition forthe oblique UAT experiments. In order to be able to compare theUAT results with CT, additional eighty eight experimaents werealso considered for CT experiments with similar condition as inUAT; a total of 176 tests were thus designed and implemented.The order of experiments was designed so that the need to changeof machining setup to keep minimum.
It should be noted that nine parameters are involved in thetheoretical model developed by the authors for oblique UAT [2].
Fig. 7. The hardness of lateral surface versus the cutting velocity VC (or rotational
speed N) in UAT and CT for different values of feed rate.
Fig. 8. The effect of different machining parameters on depth
UAT CT
Fig. 6. Optical microscope image of lateral surface for UAT and CT; VC¼0.18 m/s,
Vf¼0.4 mm/rev.
It was not possible to directly measure all of them including thegeometrical parameters. However, the force parameters wereconverted into 3D cutting forces in the radial, tangential andaxial directions in xyz coordinate system, i.e., RTX
ðtÞ, RTYðtÞ and
RTZðtÞ, by using Eq. (5-A) in part II [2]. These forces could then be
directly compared with the averaged cutting forces obtainedexperimentally by dynamometer. The theoretical model is thusvalidated directly for the cutting forces and indirectly for thegeometrical parameters through comparing the converted forceparameters to the average cutting force components, RTX
ðtÞ, RTYðtÞ
and RTZðtÞ, measured by the dynamometer.
3. Results and discussion
3.1. Cutting kinematics
Fig. 6 shows the effect of ultrasonic vibration on the lateralsurface. As can be seen from this figure, ultrasonic vibrationcreates periodic pressing marks on the lateral surface which isconsistent with the predictions of the kinematics analysis inpart I.
The results of surface hardness measurements for the lateralsurface have been plotted in Fig. 7 for UAT and CT.
It is obvious from Fig. 7 that in low cutting velocities, theultrasonic vibration considerably increases the hardness of thelateral surface, qualitatively confirming the mechanism proposedfor the relative movement of the cutting tool and workpiece andthe predicted results shown in Fig. 8-a (adopted from part I [1]).
In fact, as is clear from this figure, wider areas of pressed regionsare encountered at low cutting velocities. This increases the share ofpressed areas in the average values of hardness measurementswhich means higher values for the surface hardness. It is alsoobvious from Fig. 7 that the hardness decreases in UAT with anincrease in the cutting velocity whereas it is rather constant in CT.This means that with an increased cutting velocity the share of thepressed regions in the average hardness value is decreased, which isin agreement with the results of kinematics model in Fig. 8-aindicating that the width of pressed regions decreases.
It can also be concluded from Fig. 7 that the hardness in UATapproaches that in CT as the cutting velocity increases. In fact, thisimplies a rapid decrease of ultrasonic effect at higher cuttingvelocities as the latter approaches its critical value in UAT.
The variation of hardness of lateral surface with feed rate isillustrated in Fig. 9. It is clear from this figure that the hardness of
and width of pressed regions (adopted from part I [1]).
Fig. 11. The effect of ultrasonic vibration amplitude on the width and depth of
pressed regions; VC¼0.18 m/s, Vf¼0.4 mm/rev (adopted from Part I [1]).
Table 4Johnson–Cook constants for material Al 2024 [16,17].
A (MPa) B (MPa) N C m Troom Tmelt
352 440 0.42 0.0083 1 294 775
Table 3Chemical composition of Al 2024 [5].
Si Fe Cu Mn Mg Cr Zn Ti Al
0.5%
max
0.5
max
3.8–
4.9%
0.3–
0.9%
1.2–
1.8%
0.1%
max
0.25%
max
0.15%
max
93.50%
Fig. 9. The hardness of lateral surface versus feed rate in UAT and CT for different
values of cutting velocity VC.
Fig. 10. The hardness of lateral surface versus the vibration amplitude in UAT.
H. Razavi et al. / International Journal of Mechanical Sciences 63 (2012) 26–3630
lateral surface increases with an increase in the feed rate. This ismore considerable in UAT process than in CT and is also inagreement with the predicted results illustrated in Fig. 8-b whichimplies an increase in depth with an increase in feed rate. In fact,an increase in the pressing depth causes the increase of hardness.
Several UAT experiments were also carried out at low and highcutting velocities and feed rates by changing the vibrationamplitude. The results are presented in Fig. 10.
It can be observed from this figure that as vibration amplitudeincreases, the hardness of the lateral surface also increases which isdue to the increased influence of the ultrasonic vibration. Anincrease in the vibration amplitude results in a longer forward andbackward motion of the cutting tool leading to increased width ofpressed regions (Fig. 11, adopted from part I [1]) which in turncauses increase of the average hardness value as explained above.
3.2. Cutting dynamics
In order to theoretically estimate the parameters of thedynamic behavior in oblique cutting, a system of nine equationsare solved for nine different unknowns including the shear angleand the forces acting on the rake face and on the chip surface [2].The ultimate effect of these parameters can be evidenced from thechange of the cutting force components. Therefore, the theoreticalcutting forces are compared with the experimental cutting forcesmeasured by dynamometer as a means of validating the theoreticalmodel. The influence of various oblique cutting parameters on thecutting force components is also experimentally investigated.
It should be noted that the relations governing oblique CT areobtained by substituting a¼0 and f¼0 in oblique UAT relations.
It is assumed that the coefficient of friction between the chipand the tool rake face is constant and denoted by mmean (average
friction angle by bmean). This is considered to be an average valueof the sticking and sliding friction coefficients based on the modelproposed by Zorev [4]. For an explanation about the friction angle,Fig. 2 in part II [2] may be consulted. The influence of theultrasonic vibration on the mechanical properties of materialhas been ignored.
3.2.1. Material properties
The chemical composition of Al 2024 is presented in Table 3 [5].Generally, the deformation of the workpiece material in a
machining process occurs under the following conditions; largestrains in the range of 2–4 for cutting processes [5,6]; high strainrates in the range of 103–106 s�1 for CT and UAT [6–11]; and hightemperatures up to 600 K being developed due to the plastic workin the cutting zone and also because of the friction existing betweenthe cutting tool, workpiece and chip [9,10,12,12,13]. The flow stressof the material depends mainly on the above-mentioned threefactors in addition to the microstructure, although the degree ofdependence differs for different materials. In the present investiga-tion, the distribution of the shear flow stress on the shear plane isassumed to be uniform and constant representing a mean value forranges of the temperature and deformation rate. The Johnson–Cookformula is employed to estimate the shear stress for CT and UATprocesses, as follows [14]:
s¼ AþBen� �� 1þCln
_e_e0
!" #� 1�
T�Troom
Tmelt�Troom
� �m� �ð1Þ
The coefficients of Johnson–Cook’s model are presented for Al2024 in Table 4. These coefficients are determined from Split
Table 5Theoretical and experimental cutting forces and relative error in oblique cutting.
Run Parameters Theoretical results Experimental results Relative error
VC (m/s) N (rev/min) Vf (mm/rev) b (mm) i (deg) Kr (deg) a (mm) CT-UAT RT X (N) RT Y (N) RT Z (N) RT X (N) RT Y (N) RT Z (N) RT X (N) RT Y (N) RT Z (N)
1 0.8 80 1 1 15 60 6 UAT 218.6 683.2 139.2 322.94 980.95 228.60 32.31 30.35 39.11
2 0.8 80 1 1 15 60 6 CT 218.6 683.2 139.2 342.48 1097.82 240.97 36.17 37.77 42.23
3 0.8 80 0.4 1 15 75 6 UAT 70.06 273.3 76.43 112.15 416.92 125.32 37.53 34.45 39.01
4 0.8 80 0.4 1 15 75 6 CT 70.06 273.3 76.43 108.39 421.06 122.59 35.36 35.09 37.65
5 1.3 131 0.05 1 15 90 6 UAT 5.987 35.16 11.49 2.38 58.46 20.13 �151.53 39.86 42.91
6 1.3 131 0.05 1 15 90 6 CT 5.987 35.16 11.49 1.78 64.88 22.79 �236.68 45.80 49.57
7 0.53 53 0.4 2 15 75 6 UAT 86.2 336.2 94.04 142.52 538.19 138.41 39.52 37.53 32.06
8 0.53 53 0.4 2 15 75 6 CT 140.1 545.5 152.9 227.34 685.81 227.40 38.38 20.46 32.76
9 0.8 80 0.05 2 15 90 6 UAT 11.97 68.32 22.99 5.96 104.51 44.33 �100.79 34.63 48.14
10 0.8 80 0.05 2 15 90 6 CT 11.97 68.32 22.99 13.88 117.88 55.36 13.78 42.04 58.47
11 2.1 211 0.4 2 15 90 6 UAT 95.79 546.5 183.9 159.32 750.46 283.94 39.88 27.18 35.23
12 2.1 211 0.4 2 15 90 6 CT 95.79 546.5 183.9 151.33 763.39 287.99 36.70 28.41 36.14
13 1.3 131 0.4 1 30 60 6 UAT 130.7 272 23.76 178.28 403.00 35.29 26.69 32.51 32.67
14 1.3 131 0.4 1 30 60 6 CT 130.7 272 23.76 202.27 458.05 45.99 35.38 40.62 48.34
15 2.1 211 1 1 30 60 6 UAT 326.6 680.1 59.39 461.86 1053.21 99.57 29.29 35.43 40.35
16 2.1 211 1 1 30 60 6 CT 326.6 680.1 59.39 481.16 1043.69 99.54 32.12 34.84 40.34
17 2.1 211 0.05 1 30 90 6 UAT 12.66 34.01 10.74 �8.75 54.82 7.18 244.61 37.96 �49.51
18 2.1 211 0.05 1 30 90 6 CT 12.66 34.01 10.74 20.92 66.00 15.81 39.48 48.47 32.09
19 0.8 80 0.4 1 30 90 6 UAT 101.3 272 85.9 171.74 421.91 131.79 41.01 35.53 34.82
20 0.8 80 0.4 1 30 90 6 CT 101.3 272 85.9 161.04 437.95 140.51 37.09 37.89 38.87
21 0.8 80 0.05 2 30 60 6 UAT 32.66 68.01 5.93 42.41 114.11 4.35 22.98 40.40 �36.38
22 0.8 80 0.05 2 30 60 6 CT 32.66 68.01 5.93 51.60 103.70 21.11 36.70 34.42 71.90
23 0.53 53 1 2 30 60 6 UAT 402 837.1 73.1 572.21 1201.88 123.21 29.75 30.35 40.67
24 0.53 53 1 2 30 60 6 CT 653.3 1360 118.8 876.87 1712.19 197.36 25.50 20.57 39.81
25 1.3 131 0.05 2 30 75 6 UAT 30.01 68.01 14.19 42.80 104.62 17.80 29.88 34.99 20.30
26 1.3 131 0.05 2 30 75 6 CT 30.01 68.01 14.19 46.92 115.19 39.45 36.03 40.96 64.03
27 2.1 211 0.4 2 30 75 6 UAT 240.1 544.1 113.5 347.78 795.32 185.35 30.96 31.59 38.76
28 2.1 211 0.4 2 30 75 6 CT 240.1 544.1 113.5 371.27 774.93 200.58 35.33 29.79 43.41
29 2.1 211 0.05 1 30 60 11 UAT 16.33 34.01 2.97 �15.68 64.04 14.76 204.14 46.90 79.88
30 2.1 211 0.05 1 30 60 11 CT 16.33 34.01 2.97 42.73 79.58 26.36 61.78 57.26 88.73
31 0.8 80 0.4 1 30 75 11 UAT 63.44 143.8 30 96.49 235.76 46.58 34.25 39.01 35.59
32 0.8 80 0.4 1 30 75 11 CT 120.1 272 56.77 186.58 413.20 94.70 35.63 34.17 40.06
33 1.3 131 0.4 1 30 90 11 UAT 86.29 231.8 73.19 133.53 222.34 101.47 35.38 �4.25 27.87
34 1.3 131 0.4 1 30 90 11 CT 101.3 272 85.9 158.50 416.95 129.83 36.09 34.76 33.84
35 0.53 53 1 1 30 90 11 UAT 101.3 272.1 85.9 132.56 399.99 102.68 23.58 31.97 16.34
36 0.53 53 1 1 30 90 11 CT 253.2 680.1 214.8 317.55 997.62 276.20 20.26 31.83 22.23
37 0.53 53 0.4 2 30 60 11 UAT 104.5 217.7 19.01 160.85 341.44 36.53 35.03 36.24 47.97
38 0.53 53 0.4 2 30 60 11 CT 261.3 544.1 47.51 398.59 784.11 78.43 34.44 30.61 39.42
39 0.8 80 1 2 30 75 11 UAT 317.2 718.9 150 469.85 1046.76 245.23 32.49 31.32 38.83
40 0.8 80 1 2 30 75 11 CT 600.3 1360 283.8 809.50 1873.83 427.43 25.84 27.42 33.60
41 0.8 80 0.05 2 30 90 11 UAT 13.36 35.94 11.35 17.24 60.76 4.67 22.52 40.85 �142.9
42 0.8 80 0.05 2 30 90 11 CT 25.32 68.01 21.48 40.35 109.08 20.16 37.24 37.65 �6.53
43 0.8 80 0.05 1 15 75 11 CT 8.758 34.16 9.554 7.86 60.52 5.21 �11.41 43.55 �83.35
44 1.3 131 1 1 15 75 11 UAT 149.2 582.1 162.8 244.03 801.06 252.85 38.86 27.33 35.61
45 1.3 131 1 1 15 75 11 CT 175.2 683.2 191.1 261.70 841.48 287.24 33.05 18.81 33.47
46 0.53 53 0.05 1 15 90 11 UAT 2.392 13.65 4.592 �6.38 38.29 27.46 137.48 64.35 83.28
47 0.8 80 1 1 15 90 11 UAT 63.22 360.8 121.4 99.13 557.20 175.42 36.23 35.25 30.79
48 0.8 80 1 1 15 90 11 CT 119.7 683.2 229.9 188.79 930.46 330.39 36.60 26.57 30.42
49 0.8 80 0.4 2 15 60 11 UAT 92.36 288.6 58.82 145.07 433.88 98.34 36.33 33.48 40.19
50 0.8 80 0.4 2 15 60 11 CT 174.9 546.5 111.4 264.44 775.51 185.71 33.86 29.53 40.01
51 0.53 53 1 2 15 60 11 UAT 174.7 545.9 111.2 252.67 763.99 184.85 30.86 28.55 39.84
52 0.53 53 1 2 15 60 11 CT 437.3 1366 278.5 614.53 1883.40 421.55 28.84 27.47 33.93
53 2.1 211 0.05 2 15 75 11 CT 17.52 68.32 19.11 23.90 115.67 18.15 26.69 40.94 �5.29
54 2.1 211 1 2 15 75 11 UAT 350.3 1366 382.1 479.15 1698.63 688.77 26.89 19.58 44.52
55 2.1 211 1 2 15 75 11 CT 350.3 1366 382.1 511.05 1789.89 536.78 31.45 23.68 28.82
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Table 5 (continued )
Run Parameters Theoretical results Experimental results Relative error
VC (m/s) N (rev/min) Vf (mm/rev) b (mm) i (deg) Kr (deg) a (mm) CT-UAT RT X (N) RT Y (N) RT Z (N) RT X (N) RT Y (N) RT Z (N) RT X (N) RT Y (N) RT Z (N)
56 1.3 131 1 2 15 90 11 UAT 204 1164 391.7 322.16 1582.20 531.08 36.68 26.43 26.24
57 1.3 131 1 2 15 90 11 CT 239.5 1366 459.8 369.34 1796.34 750.81 35.15 23.96 38.76
58 1.3 131 1 1 15 60 16 UAT 126.2 394.4 80.38 192.96 585.33 135.22 34.60 32.62 40.55
59 1.3 131 1 1 15 60 16 CT 218.6 683.2 139.2 342.20 999.38 228.41 36.12 31.64 39.06
60 0.53 53 0.05 1 15 75 16 UAT 2.775 10.83 3.028 �4.51 19.13 7.72 161.55 43.40 60.78
61 0.53 53 0.05 1 15 75 16 CT 8.758 34.16 9.554 6.40 73.13 4.24 �36.83 53.29 �125.3
62 1.3 131 0.4 1 15 90 16 UAT 27.65 157.8 53.09 43.14 258.85 93.97 35.90 39.04 43.50
63 1.3 131 0.4 1 15 90 16 CT 47.89 273.3 91.96 73.76 417.56 149.61 35.08 34.55 38.53
64 0.53 53 0.05 2 15 60 16 UAT 6.928 21.65 4.412 3.55 55.28 18.66 �95.33 60.83 76.36
65 0.53 53 0.05 2 15 60 16 CT 21.86 68.32 13.92 31.54 110.81 22.78 30.68 38.34 38.90
66 1.3 131 0.05 2 15 60 16 UAT 12.62 39.44 8.038 21.48 63.83 17.69 41.24 38.21 54.56
67 1.3 131 0.05 2 15 60 16 CT 21.86 68.32 13.92 31.63 123.39 27.57 30.89 44.63 49.51
68 0.8 80 1 2 15 75 16 UAT 143 557.8 156 213.13 819.96 245.73 32.91 31.97 36.52
69 0.8 80 1 2 15 75 16 CT 350.3 1366 382.1 509.74 1788.78 540.17 31.28 23.64 29.26
70 2.1 211 0.4 2 15 90 16 UAT 95.79 546.5 183.9 156.51 746.91 279.88 38.80 26.83 34.29
71 2.1 211 0.4 2 15 90 16 CT 95.79 546.5 183.9 156.08 788.75 300.04 38.63 30.71 38.71
72 0.8 80 1 1 30 60 16 UAT 133.6 278.2 24.29 202.96 420.68 77.12 34.17 33.87 68.50
73 0.8 80 1 1 30 60 16 CT 326.6 680.1 59.39 481.82 1021.24 99.38 32.22 33.40 40.24
74 0.53 53 0.05 1 30 75 16 UAT 4.766 10.8 2.254 �2.12 17.46 7.32 324.37 38.16 69.20
75 0.53 53 0.05 1 30 75 16 CT 15.01 34.01 7.096 26.12 52.12 18.98 42.54 34.75 62.62
76 1.3 131 1 1 30 75 16 UAT 173.4 393 81.99 258.30 601.05 138.10 32.87 34.61 40.63
77 1.3 131 1 1 30 75 16 CT 300.1 680.1 141.9 448.86 994.89 233.16 33.14 31.64 39.14
78 1.3 131 0.05 1 30 90 16 UAT 7.314 19.65 6.204 6.93 34.24 6.82 �5.49 42.61 9.08
79 1.3 131 0.05 1 30 90 16 CT 12.66 34.01 10.74 23.31 60.50 4.25 45.68 43.78 �152.5
80 0.53 53 0.4 1 30 90 16 UAT 32.16 86.43 27.28 51.92 149.84 42.09 38.06 42.32 35.18
81 0.53 53 0.4 1 30 90 16 CT 101.3 272 85.9 154.92 420.44 156.56 34.61 35.31 45.13
82 1.3 131 0.4 2 30 60 16 UAT 151 314.4 27.45 220.81 465.95 42.11 31.62 32.53 34.81
83 1.3 131 0.4 2 30 60 16 CT 261.3 544.1 47.51 399.25 773.15 78.89 34.55 29.63 39.78
84 2.1 211 1 2 30 90 16 UAT 506.4 1360 429.5 672.48 1740.52 605.28 24.70 21.86 29.04
85 2.1 211 1 2 30 90 16 CT 506.4 1360 429.5 695.80 1792.69 704.60 27.22 24.14 39.04
86 1.3 131 0.05 1 45 75 6 UAT 23.2 34.64 4.027 35.98 57.02 12.59 35.51 39.25 68.02
87 1.3 131 0.05 1 45 75 6 CT 23.2 34.64 4.027 42.44 70.69 18.44 45.34 51.00 78.16
88 0.53 53 0.4 1 45 90 6 UAT 105.2 170.6 48.73 164.91 261.93 80.95 36.21 34.87 39.80
89 0.53 53 0.4 1 45 90 6 CT 171 277.1 79.16 248.08 419.56 132.81 31.07 33.95 40.40
90 2.1 211 1 1 45 90 6 UAT 427.4 692.9 197.9 573.23 1024.52 313.72 25.44 32.37 36.92
91 2.1 211 1 1 45 90 6 CT 427.4 692.9 197.9 602.71 979.03 354.96 29.09 29.23 44.25
92 2.1 211 0.05 2 45 60 6 UAT 46.91 69.29 �4.23 64.15 114.75 �17.76 26.88 39.62 76.18
93 2.1 211 0.05 2 45 60 6 CT 46.91 69.29 �4.23 77.21 118.13 �6.65 39.24 41.34 36.35
94 0.8 80 1 2 45 60 6 UAT 983.2 1386 �84.6 1352.611 1803.534 �188.529 27.31 23.15 55.11
95 0.8 80 1 2 45 60 6 CT 983.2 1386 �84.6 1320.086 1864.707 �115.842 25.52 25.67 26.94
96 0.53 53 1 2 45 75 6 UAT 571.3 853 99.14 773.86 1157.32 158.30 26.18 26.29 37.37
97 0.53 53 1 2 45 75 6 CT 928.1 1386 161.1 1264.24 1788.20 251.21 26.59 22.49 35.87
98 1.3 131 0.4 2 45 90 6 UAT 341.9 554.3 158.3 480.79 806.22 247.39 28.89 31.25 36.01
99 1.3 131 0.4 2 45 90 6 CT 341.9 554.3 158.3 515.72 812.45 269.38 33.70 31.77 41.24
100 0.53 53 0.05 1 45 60 11 UAT 9.396 13.88 �0.85 7.77 48.57 �11.27 �20.90 71.42 92.47
101 0.53 53 0.05 1 45 60 11 CT 23.46 34.64 �2.12 34.51 55.36 �13.27 32.02 37.43 84.06
102 2.1 211 0.4 1 45 60 11 UAT 187.6 277.1 �16.9 276.12 420.21 �28.16 32.06 34.06 39.88
103 2.1 211 0.4 1 45 60 11 CT 187.6 277.1 �16.9 285.38 394.87 �42.63 34.26 29.82 60.29
104 1.3 131 1 1 45 60 11 UAT 399.7 590.4 �36.1 569.90 902.77 �82.43 29.87 34.60 56.25
105 1.3 131 1 1 45 60 11 CT 469.1 692.9 �42.3 633.88 1017.48 �95.05 26.00 31.90 55.48
106 0.53 53 0.4 1 45 75 11 UAT 74.36 111 12.9 118.61 176.33 15.43 37.31 37.05 16.39
107 0.53 53 0.4 1 45 75 11 CT 185.6 277.1 32.21 266.52 415.61 51.05 30.36 33.33 36.91
108 0.8 80 0.4 2 45 90 11 UAT 180.9 293.2 83.73 264.34 444.91 140.51 31.56 34.10 40.41
109 0.8 80 0.4 2 45 90 11 CT 341.9 554.3 158.3 555.95 840.44 249.37 38.50 34.05 36.52
110 1.3 131 0.05 1 45 60 16 UAT 13.57 20.04 �1.23 16.13 70.78 �12.34 15.88 71.69 90.08
111 1.3 131 0.05 1 45 60 16 CT 23.46 34.64 �2.12 34.88 86.06 �13.74 32.74 59.75 84.61
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112 0.8 80 0.4 1 45 75 16 UAT 76.07 113.6 13.19 120.25 179.37 15.77 36.74 36.67 16.38
113 0.8 80 0.4 1 45 75 16 CT 185.6 277.1 32.21 279.05 405.46 62.81 33.49 31.66 48.72
114 2.1 211 1 1 45 75 16 UAT 464.1 692.9 80.53 606.77 958.21 135.62 23.51 27.69 40.62
115 2.1 211 1 1 45 75 16 CT 464.1 692.9 80.53 645.58 1017.70 151.35 28.11 31.91 46.79
116 2.1 211 0.05 1 45 90 16 UAT 21.37 34.64 9.895 51.68 50.72 4.64 58.65 31.71 �113.2
117 2.1 211 0.05 1 45 90 16 CT 21.37 34.64 9.895 73.98 59.31 5.29 71.12 41.59 �87.04
118 0.53 53 0.4 2 45 60 16 UAT 119.5 176.6 �10.8 186.21 283.70 �30.33 35.83 37.75 64.40
119 0.53 53 0.4 2 45 60 16 CT 375.3 554.3 �33.9 540.33 785.58 �76.97 30.54 29.44 56.02
120 0.53 53 0.05 2 45 75 16 UAT 14.78 22.08 2.562 18.19 36.98 1.42 18.75 40.30 �80.46
121 0.53 53 0.05 2 45 75 16 CT 46.41 69.29 8.054 75.38 115.80 17.86 38.43 40.17 54.89
122 1.3 131 0.05 2 45 75 16 UAT 26.84 40.08 4.656 40.03 65.74 17.21 32.95 39.04 72.95
123 1.3 131 0.05 2 45 75 16 CT 46.41 69.29 8.054 75.99 107.96 25.85 38.92 35.82 68.84
124 2.1 211 0.4 2 45 75 16 UAT 371.3 554.3 64.43 534.41 771.26 86.75 30.52 28.13 25.73
125 2.1 211 0.4 2 45 75 16 CT 371.3 554.3 64.43 546.79 800.93 95.33 32.09 30.79 32.41
126 0.8 80 1 2 45 90 16 UAT 350.3 568 162.1 512.97 800.93 237.75 31.71 29.08 31.82
127 0.8 80 1 2 45 90 16 CT 854.8 1386 395.8 1155.37 1763.60 545.56 26.01 21.41 27.45
128 1.3 131 1 2 45 90 16 UAT 494.4 801.6 228.9 683.36 1130.85 328.38 27.65 29.12 30.29
129 1.3 131 1 2 45 90 16 CT 854.8 1386 395.8 1155.91 1858.16 574.82 26.05 25.41 31.14
130 2.1 211 0.4 2 0 60 16 UAT 94.53 549.1 163.7 139.46 839.35 265.04 32.22 34.58 38.23
131 2.1 211 0.4 2 0 60 16 CT 94.53 549.1 163.7 158.65 1023.63 304.46 40.42 46.36 46.23
132 0.53 53 1 2 0 75 16 UAT 38.73 434.6 144.5 61.31 666.72 229.38 36.83 34.81 37.00
133 0.53 53 1 2 0 75 16 CT 122.3 1373 456.5 186.67 1746.17 620.90 34.48 21.37 26.48
134 0.8 80 0.4 2 0 90 16 UAT 3E-06 224 77.12 �11.05 268.09 125.64 100.00 16.45 38.62
135 0.8 80 0.4 2 0 90 16 CT 6E-06 549.1 189.1 5.40 799.69 325.97 100.00 31.34 41.99
136 0.8 80 0.05 1 0 90 16 UAT 2E-07 14 4.82 �9.39 32.77 6.13 100.00 57.28 21.36
137 0.8 80 0.05 1 0 90 16 CT 4E-07 34.2 11.82 �2.69 62.42 23.90 100.00 45.21 50.54
138 2.1 211 1 1 0 90 16 UAT 8E-06 686.3 236.3 �11.32 978.47 361.43 100.00 29.86 34.62
139 2.1 211 1 1 0 90 16 CT 8E-06 686.3 236.3 �4.67 1015.09 399.04 100.00 32.39 40.78
140 0.53 53 0.4 1 0 60 11 UAT 18.87 109.6 32.69 26.70 183.00 51.50 29.32 40.11 36.53
141 0.53 53 0.4 1 0 60 11 CT 47.26 274.5 81.86 78.32 415.71 137.31 39.66 33.97 40.38
142 2.1 211 0.4 1 0 75 11 UAT 24.47 274.5 91.31 13.67 408.59 121.89 �78.94 32.82 25.09
143 2.1 211 0.4 1 0 75 11 CT 24.47 274.5 91.31 36.61 447.16 153.51 33.16 38.61 40.52
144 0.8 80 0.05 2 0 60 11 UAT 6.238 36.23 10.8 2.24 58.49 26.74 �178.81 38.06 59.62
145 0.8 80 0.05 2 0 60 11 CT 11.82 68.63 20.47 13.61 114.96 29.75 13.14 40.30 31.19
146 1.3 131 0.05 2 0 60 11 UAT 10.07 58.47 17.44 9.42 98.03 23.72 �6.91 40.36 26.49
147 1.3 131 0.05 2 0 60 11 CT 11.82 68.63 20.47 19.05 115.16 61.29 37.94 40.41 66.60
148 2.1 211 1 2 0 60 11 UAT 236.3 1373 409.3 342.87 1747.38 580.89 31.08 21.43 29.54
149 2.1 211 1 2 0 60 11 CT 236.3 1373 409.3 365.86 1848.11 791.22 35.41 25.71 48.27
150 0.53 53 0.05 2 0 75 11 UAT 2.442 27.4 9.115 �5.73 64.96 4.91 142.63 57.82 �85.53
151 1.3 131 0.4 2 0 75 11 UAT 41.69 467.8 155.6 68.10 660.86 244.41 38.79 29.21 36.34
152 1.3 131 0.4 2 0 75 11 CT 48.93 549.1 182.6 81.44 826.26 282.20 39.92 33.54 35.29
153 2.1 211 0.05 2 0 90 11 UAT 8E-07 68.63 23.63 �9.61 117.38 31.13 100.00 41.53 24.08
154 2.1 211 0.05 2 0 90 11 CT 8E-07 68.63 23.63 �9.64 118.27 54.77 100.00 41.97 56.86
155 2.1 211 0.05 1 0 60 6 UAT 5.908 34.32 10.23 �35.40 74.45 9.93 116.69 53.90 �2.97
156 2.1 211 0.05 1 0 60 6 CT 5.908 34.32 10.23 1.46 62.06 20.77 �305.39 44.70 50.75
157 0.8 80 0.4 1 0 60 6 UAT 47.26 274.5 81.86 66.96 431.36 137.72 29.42 36.36 40.56
158 0.8 80 0.4 1 0 60 6 CT 47.26 274.5 81.86 78.21 412.71 150.38 39.57 33.49 45.57
159 0.53 53 1 1 0 60 6 UAT 72.68 422.2 125.9 114.56 614.04 208.69 36.55 31.24 39.67
160 0.53 53 1 1 0 60 6 CT 118.2 686.3 204.7 187.19 1060.92 323.06 36.85 35.31 36.64
161 0.53 53 0.4 1 0 75 6 UAT 15.05 168.9 56.17 18.97 272.55 93.80 20.66 38.03 40.12
162 0.53 53 0.4 1 0 75 6 CT 24.47 274.5 91.31 36.49 416.35 153.26 32.93 34.07 40.42
163 0.8 80 1 1 0 75 6 UAT 61.16 686.3 228.3 83.64 984.71 355.18 26.87 30.30 35.72
164 0.8 80 1 1 0 75 6 CT 61.16 686.3 228.3 93.22 1050.87 398.32 34.39 34.69 42.68
165 1.3 131 1 1 0 90 6 UAT 8E-06 686.3 236.3 �32.69 960.58 476.31 100.00 28.55 50.39
166 1.3 131 1 1 0 90 6 CT 8E-06 686.3 236.3 �9.22 1015.75 366.67 100.00 32.43 35.56
167 1.3 131 1 2 0 60 6 UAT 236.3 1373 409.3 356.4995 1761.195 580.9463 33.72 22.04 29.55
168 1.3 131 1 2 0 60 6 CT 236.3 1373 409.3 461.2863 1878.881 676.3829 48.77 26.92 39.49
169 2.1 211 1 2 0 75 6 UAT 122.3 1373 456.5 178.9595 1809.364 627.8046 31.66 24.12 27.29
170 2.1 211 1 2 0 75 6 CT 122.3 1373 456.5 192.7465 1717.929 708.1153 36.55 20.08 35.53
171 0.53 53 0.05 2 0 90 6 UAT 5E-07 42.22 14.54 �13.6156 73.99025 18.98719 100.00 42.94 23.42
172 0.53 53 0.05 2 0 90 6 CT 8E-07 68.63 23.63 �16.5574 110.1887 27.10102 100.00 37.72 12.81
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H. Razavi et al. / International Journal of Mechanical Sciences 63 (2012) 26–3634
Hopkinson pressure or torsion bar (SHPB) and machining tests forhigh strains, high strain rates and high temperature conditions[15,16].
The following ranges were considered for the deformationparameters: the equivalent strain e¼ 2�4, the equivalent strainrate _e¼ 103
�106 s�1 and the temperature T ¼ 300�500 1C. Anaverage value was then calculated for the flow stress s consider-ing all the eight possible combinations of the extreme values ofstrain, strain rate and temperature. By applying Huber-von missescriterion, the value of equivalent shear stress t is calculated asfollows:
t¼ s=ffiffiffi3p¼Meanðs1 � � �s8Þ=
ffiffiffi3p¼ 387:4=
ffiffiffi3pffi224 MPa ð2Þ
In order that the theoretical results to be comparable with theexperimental ones, the theoretical cutting forces were calculatedfor 176 cutting conditions designed by the DOE algorithm, byassuming 224 MPa for the shear stress and 19 deg for the frictionangle. The chosen friction angle is in agreement with the relationb¼19.1þ0.29gn (b¼friction angle, gn¼0 in the present paper)presented by Budak and Altintas [17] and Shamoto and Altintas[18]. The chosen friction angle is also almost equal to themaximum coefficient of friction, 0.35 (tan�1ð0:19Þ), obtained byJamshidi [19]).
It should be noted that because the dynamometer’s samplingrate (maximum 3 kHz) is several times less than ultrasonicvibration frequency (about 20 kHz), so the measurement of theactual (instantaneous) forces in each cycle of vibration is impos-sible and the reported results are actually average cutting forces.
In Table 5, the theoretical and experimental results for thecutting forces and their relative error (Relative error¼[(Exp Force–
Theo Force)/Exp Force]�100) are presented.As it can be seen from Table 5, in large values of cutting forces
(in CT and UAT processes) the relative error between the theore-tical and experimental results is about 20–25%. For cutting forcesof lower values down to 50 N, the maximum error reaches up toabout 49%. These ranges of error are acceptable for machiningprocesses. At small values of cutting forces (smaller than 50 N),the relative errors are large and in some cases are more than100%. The reason can be attributed to the constant systematicerrors persisting during the experiments. The most importantsystematic error is the zero error. This was clearly evidenced forthe dynamometer which showed a constant value for the cuttingforces even in the idle running of the machine. The dynamometershowed few Newton for the cutting forces even when nomachining operation was conducted. This order of error iscomparable in magnitude to small cutting forces and the relativeerror would be evidenced at high magnification. However, theabsolute errors (Absolute error¼(Exp Force–Theo Force)) at smallcutting forces are small (about few Newton).
The theoretical and experimental cutting forces for differentparameters including amplitude a, cutting velocity VC, inclinationangle i, tool cutting edge angle Kr and feed rateVf are illustrated inFig. 12.
It is evident that the cutting forces in UAT decrease by increasingthe vibration amplitude (Fig. 12-a) and increase by increasing thecutting velocity (Fig. 12-b). The model presented in part II showsthat the changes in cutting forces in CT are insignificant as thematerial softening effect is not taken into consideration when thethin shear plane theory is employed; the experimental resultsindicate moderate decrease in the cutting forces.
By increasing the inclination angle in oblique CT and UAT(Fig. 12-c), the radial cutting force would increase and the axialcutting force would decrease, while the main cutting force almostremains constant. This trend is also observed in the experimentalresults. As the tool cutting edge angle increases in oblique UATand CT processes (Fig. 12-d), the radial cutting force decreases
and the axial cutting force increases, but the principle cuttingforce remains almost constant. This behavior can also be almostobserved in experimental results.
As the feed rate increases in oblique UAT and CT processes(Fig. 12-e), the cutting forces also increase. This is again evidencedin the experimental results.
In order to better compare the cutting forces in UAT and CT,the ratio between the resultant cutting forces created in UAT andCT is denoted by a cutting force ratio, l ðl¼ RTUAT=RTCT Þ. A lowervalue for the cutting force ratio implies that the vibration cuttingcan be implemented more effectively. The vibration cutting losesits benefits and changes to a CT process at l¼1. The variations ofthe cutting force ratio against different parameters are illustratedin Fig. 13. Both the theoretical and experimental results in thisfigure indicate that the cutting force ratio is highly influenced bythe changes of vibration amplitude and cutting speed. A decreaseup to about seventy percent is observed in the cutting force ratio atlarge amplitudes and low cutting speeds. The intermittent vibrationcutting changes to a continuous operation at low vibration ampli-tudes and high cutting speeds (l¼1) as is clear from the figure.According to Fig. 13, the experimental results indicate that thecutting force ratio slightly increases with increasing feed rate; sothe advantages of the vibration cutting process is expected to tenddownwards. This is due to the increase of the burnishing effectcaused by the cutting tool motion (Fig. 8-b and Fig. 9). It is also clearfrom Fig. 13 that the inclination angle and the tool cutting edgeangle do not have considerable effect on the cutting force ratio. Inother words, these parameters cannot be employed to increase theeffectiveness of vibration cutting.
It is evident from Fig. 13 that there is a close agreementbetween the theoretical and experimental results for l.
4. Conclusion
The theoretical models already developed and presented inearlier works for the kinematics and dynamics analysis of theultrasonic vibration assisted oblique turning (oblique UAT) havebeen experimentally verified in the present study.
The vibrating motion of the cutting tool caused a toothedpattern to be left on the lateral surface of the workpiece andalso the increase of the surface hardness in UAT experiments.The average hardness of the lateral surface increased when thecutting speed decreased. The increase of the average hardnesswas also evidenced when the feed rate or the vibration amplitudeincreased. The experiments also confirmed that the hardness ofthe lateral surface in UAT approaches to those in CT at highercutting velocities. These evidences confirms the theoretical modelpredicting the kinematic behavior of the vibration cutting processand that the cutting tool presses against the lateral surface of theworkpiece during its withdrawl and forward motions from andtwards the workpiece, respectively, occurring in the nonmachin-ing part of the vibration cycle. It is then true that the cutting tooldoes not disengage from the workpiece during its cyclic motion.
In order to estimate the parameters of the dynamic behavior inoblique cutting, a system of nine equations are solved for ninedifferent unknowns including the shear angle and the forcesacting on the rake face and on the chip surface. The ultimateeffect of these parameters can be evidenced by the change of thecutting force components. The relative error between the theore-tical and experimental results for oblique CT and UAT cuttingforces was about 20–25% for higher cutting forces and up to about49% for medium values of cutting forces which are acceptable formachining processes. At lower cutting forces (under 50 N), theabsolute error was about few Newton.
Fig. 12. Cutting forces versus oblique UAT parameters; b¼1 mm, L¼1 mm, b¼ 19 deg, r¼2780 kg/m3, t¼224 MPa. (a) i¼30 1, Kr¼75 1, VC¼0.53m/s, Vf¼0.4mm/rev,
(b) i¼30 1, Kr¼75 1, ln¼0 1, Vf¼0.4mm/rev, a¼16 mm, (c) Kr¼75 1, VC¼0.53m/s, Vf¼0.4mm/rev, a¼16 mm, (d) i¼30 1, VC¼0.53m/s, Vf¼0.4mm/rev, a¼16 mm, and (e) i¼30 1,
Kr¼75 1, VC¼0.53m/s, a¼16 mm.
H. Razavi et al. / International Journal of Mechanical Sciences 63 (2012) 26–36 35
By increasing the inclination angle in oblique CT and UAT, theradial cutting force increased and the axial cutting force decreased,whereas the main cutting force almost remained constant. Thistrend was observed in both the theoretical and experimentalresults. The radial cutting force decreased and the axial cuttingforce increased by increasing the tool cutting edge angle in oblique
UAT and CT processes, but the principle cutting force was notinfluenced.
The theoretical analysis and experimental observations showthat the inclination angle and tool cutting edge angle do not haveany considerable effect on the cutting force ratio, but the vibrationamplitude and the cutting speed do influence this ratio. An increase
Fig. 13. Variation of the ratio between the resultant cutting forces in UAT and CT (l).
H. Razavi et al. / International Journal of Mechanical Sciences 63 (2012) 26–3636
in the vibration amplitude or a decrease in the cutting speeddecreases the cutting force ratio resulting in a more effectivevibration cutting process. The experimental results indicate thatthe cutting force ratio slightly increases with increasing feed rate;so the advantages of the vibration cutting process is expected totend downwards. This is due to the increase of the burnishing effectcaused by the cutting tool motion. There was a close agreementbetween the theoretical and experimental results for the cuttingforce ratio against different parameters.
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