a Graduate School of Mechanical Engineering, Pukyong National University, Busan, Republic of Koreab School of Mechanical Engineering, Pukyong National University, Busan, Republic of Korea
Received 22 March 2004; accepted 30 June 2004Available online 21 September 2004
he theoretical roundness profile of a hole with an arbitrary multi-lobe can be calculated. But in real experiments, the two- to sixrofile was frequently measured. The most frequently measured one is three- and five-lobe profile in experiments. With these resredicted that the reliability of proposed harmonic model has been verified theoretically and experimentally by a large number of rstimation of the hole and the experimental results that was produced by BTA drilling operation in deep hole making operation.odeling method is expected to provide desirable insights into the advanced tolerance analysis of circular hole making in BTA drillin2004 Elsevier Inc. All rights reserved.
The modeling application in BTA (boring and trepanningssociation) hole drilling is frequently found in the wide rangef industries, such as a hole drilling of machinery part, nu-lear power, oil gas, and aerospace one. The round shapeodeling of BTA drilling, considering spindle error motion
s essential for the analysis of the lobe shape and its toleranceor circular and cylindrical profile. So, roundness modelingas received substantial attention in literature. This kind ofoundness modeling is given a substantial attention in liter-ture and some papers are reviewed below. Some productsequire a high demand on quality as well as precise dimen-ion and shape tolerances. So, the experimental result abouthis quality was discussed in making a hole in SK3 alloyool steel with BTA tool under the several cutting conditions,nd compared with the theoretical modeling. Generally, the
deep hole drilling is defined as the length to diameter rabigger than 20. The BTA drilling and gun drilling are clsified in this type. Many researchers have studied abouBTA drilling. Sakuma et al.[1] studied about the profiledeep hole and its generating mechanism experimentallyresult is focused on the fields of the optimum cutting edgesign of BTA drill that generates minimum cutting force. AlSawabe et al.[2] established a relationship between thedial error motion of a spindle and the resultant part prfor turning processes after machining. Damir[3] developedan approximate harmonic model to determine the amplspectra of harmonic roundness lobe profiles in differentchining processes. Cho and Tu[4] also developed the rounness lobe modeling of machined parts for tolerance asis. They showed that the spindle error motion of a spicould result in a similar roundness part profile. For exama three-lobe spindle error motion frequency can resultthree-lobe circular profile. Their analysis, however, wasfocused on the error motions with integer multiples of sdle rotational frequencies and also considers the impor
of fractional frequency error motions and tool vibration inturning operation. Furthermore, there are many researcherswho are interested in the roundness behaviors of the hole[5–11].
In this study, a new roundness modeling of BTA drillingand its characteristics were investigated. By comparing withthe theoretical and experimental lobe results, the reliabilityof the model was proven. A more complete characteristics ofBTA drilling for the alloy tool steel and the accuracy of holewere investigated by theoretical calculation and experimentalmeasurements by performing CNC solid type BTA drilling.The BTA drill head used for drilling was depicted inFig. 1.
2. Roundness modeling of BTA drilling
A roundness profile model in BTA drilling is developedto describe the effect of spindle error motion on the round-ness profile of drilled hole.Fig. 2is a hole-generating modelof BTA drilling developed by considering the revolution ofspindle error of BTA tool center. A roundness lobe profileis the inner line of an object drilled in a given plane, andthis circular profile is defined at a plane perpendicular to theaxis of a cylinder shape. For a cylinder, the profile can be de-fined at the surface intersected by any plane passing througha hafto ousr in ac ularc tings
iont , theit
Fig. 2. Round shape lobe profile generated by the error motion of BTA tool.
instantaneous center of BTA tool with spindle error motionthat rotates in a harmonic circle.
The radius,Ra (t,θ), that is the radius of the locus rotatesat a revolution frequency ofωa , and its radius isOcOi. Thevector representation from the pointOc toOi may be writtenas follows:→
Ra(t) = x + jy = Ra(t, θ)e(ωat+φa)j (1)
wherej is imaginary unit. But,
x = Ra(t, θ)cos(ωat + φa) y = Ra(t, θ)sin(ωat + φa)
The pointTp is the fixed edge position of the BTA tooland it is caused by the revolution of the BTA tool center andworkpiece vibration with the amplitude ofRw(t), into the xdirection because of its poor fixing. InFig. 2, the distance
common center. If there is a spindle error in BTA tool sf deep hole drilling it soon becomes stable in continuotating state. And the center of the BTA tool revolvesircular or oval path of stationary state at last. But irregrossing and wandering through the center at initial rotatate may also appear.
By assuming a BTA tool motion in a stable revoluthat the trace of tool center follows circular movementsdealized harmonic model may be shown inFig. 1. In Fig. 2,he pointOc is the center of absolute coordinate andOi is the
OiT , fromOi to T, may beRl (t) and the distanceOiTp be-tweenOi andTp may be represented asRp(t). Then, the BTAtool rotates with radiusRp(t) and it has rotating frequency ofωa locating its center atOi . So the right point trace of radiusRp(t) matches the real profile of radial distanceR1(t), con-sidering the spindle error motion. Also, the movement of thepositionTp, considering in workpiece is also the movementof spindle error into x direction with its amplitude ofRw(t)and a frequency ofωw. So it is defined that the final profileof the roundness may be caused by the combination of thespindle axis error motion that is the rotating with a radiusaround the absolute center and the low stiffness workpiecevibration. These final traces of the edge,Tp, are the profilesof the roundness geometry BTA drilled. So, the equation thatrepresents this trace may be summarized as follows[7]:→
R1(t) =→
Ra(t) +→
Rp(t) (2)
= Ra(t)e(wat+φa)j + Rp(t)e(wwt+φa)j (3)
but,→
Rl (t) =→OT −
→OOi ,
→Ra(t) =
→OOi −
→OOc
→Rp(t) =
→OTp −
→OOi
R
olc lanea rofilea
bin
p sedaf file.Fm
R
singt
R
w
n
c
By using discrete form ofEq. 7, it generates frequencyof lobe profile by the natural modes offa (ωa/2π) and fw(ωw/2π) or their combination in frequency domain that areits intrinsic natural modes.
3. Analysis of predicted lobe
3.1. Half frequency effect on roundness
In hole making mechanism, there are two distinctive pat-terns in drilled profile according to the frequency offa such asone of lower frequency less than unit (fractional frequency)and the other of higher than unit. To predict former becauseof this fractional frequency error spindle motions, the feedper revolution of the BTA drilling mechanism is assumed tobe small enough so that the same plane is cut. The formerfractional half frequency generally shows the closed lobe oropen lobe in one revolution of the BTA tool. But if it ro-tates continuously then the open lobe becomes closed one inFigs. 3 and 4and almost a circle hole profiles except someexceptional condition of cycle.
Even if it is not a closed lobe, that means some phase shiftexists in one cycle, may also reveals the similar lobe profilewith little error and becomes finally circular hole when itr et cya ure.S le int thel nf off ro
The condition that some arbitrary position of BTA toutting edge coincides with the same position in same pfter one rotation may be defined as the closed lobe pnd its condition may be summarized as follows:
→R1(t) =
→R1(t + ∆t) (5)
ut,�t = 2π�
.
If the position does not coincide with the prepositionlane after one revolution it is not a closed lobe. It caunon-continuous profile. But if the valuesfa(ωa/2π) and
w(ωw/2π) be a integers, then it reveals a closed lobe prourthermore, the amplitude of trace locus of lobe profileR1(t)ay be rewritten as follows:
1(t) =∣∣∣∣
→r1(t)
∣∣∣∣ . (6)
The mode may be calculated in frequency domain by uhe discrete fourier transform as follows:
1(fk) = 1
N
N−1∑i=0
∣∣∣∣→
r1(t)
∣∣∣∣i
exp
(−j
2π
Nfki
)(7)
hereN: sampling number,fk: number of lobe (0, 1,. . .,
otates several cycles. InFigs. 3 and 4, the left figure is thrace of one revolution of BTA tool center for half frequennd the right one is the outer hole profile of the left figo, this outer shape becomes the real profile of the ho
hat workpiece drilled. For fractional frequency motion inobe profile of arbitraryfa, the lobe of workpiece vibratiorequency (fw) has the tendency of merging into the lobea as inFig. 3. Especially in even frequency (fw), the numbef lobe becomes the half offw.
Also, to predict hole profiles caused by higher frequef BTA spindle error motion, the frequency that is hig
han unit generally shows the closed or open lobe in onelution of workpiece. But if it rotates continuously the op
obe becomes closed one and also shows the frequencycteristics offa. The predicted profile of one cycle for h
requency in BTA drilling was obtained as inFigs. 3 and 4.he outer profile of the left figure reveals the right oneost case, for open lobe it shows some combination o
hifting, but shows higher radial deviation in profile compng to the lobe of fractional frequency. The frequency, suca, has a clear effect on the lobe profiles for closed lobe. Ahen a tool is vibrating at a frequency offw, both frequenies,fa andfw, decide the dominant lobe of the profile tppears in the merged lobes of the two frequencies,fa andw. These frequencies decide the dominant lobe of thele that appears in the merged lobes of the two frequenhis predicted lobe profile is shown inFig. 3 andFig. 4. Inost survey about the fractional frequency motion, the lecome closed ones and show little radial deviation inrofile of the hole comparing to that of circular lobe. Soase of low feedrate, the unit frequency offw or no move
Fig. 3. Predicted lobe profile of the BTA hole for half frequency withfa = 0.5 (Ra/Rw = 0/5). (a)fw = 1, (b) fw = 3, (c) fw = 5, (d) fw = 7, (e)fw = 9.
ment (fw = 0) does not give a great effect on lobe profiles inFig. 3a.
For this closed lobe, the multi-number frequency offwdecides the dominant lobe of the profile. This predicted holeprofile is shown inFig. 3. Note that the predicted hole profilesin this study and their corresponding tolerance error contoursare explicitly look alike. But the lobe difference of the holeis quite visible.
3.2. Multi frequency effect on roundness
Fig. 5a–u shows the predicted roundness profile usingEqs. (2)–(4). In the condition of the several values offa/fwthe predicted roundness lobe profiles is generated by BTAtool error motion and the workpiece motion. These round-ness lobe profiles include arbitrary one ton-lobe contourscorresponding to its frequencyfa and fw, respectively. In
prediction, all idealized roundness may be calculated bychanging the frequencies,fa and fw at any condition. Theresults ofFig. 5 are obtained at the condition ofRa/Rw =3/3. For the lobe condition having the extreme large orsmall value ofRa/Rw (0 or ∞) this condition may result insymmetrical lobe similar to the lobe rangingfa/fw = 1/6–6/1in Fig. 5. The value,Ra/Rw around one, may cause thebi-symmetrically merged lobe depending on the frequenciesfa andfw, respectively, as inFig. 5. But this bi-symmetricalshape may be changed into non-symmetrical lobe at differentinitial phase values ofφa andφw (Fig. 13). These results re-veal that the real frequency of the BTA tool error motion thatis caused by its poor bearing of the spindle has happened atthe most frequent value of 2–6 in frequency,fa (= ωa/2π) andfw (= ωw/2π). The two- to six-lobe contours are most fre-quently generated in real experiments. Six or more multi-loberoundness profiles have not frequently generated in real
machining. Most experimental number of lobe contoursis less than seven. Even if there are many combinationsof frequencyfa and fw, its combinations generally revealtwo- to six-lobe contours. These results mean that BTAtool error motion revolute ranging 2–6 offa or workpiece
Ff
vibration in such a condition of 2–6 offw. But the stiffnessin workpiece is stronger than that in spindle of the BTAtool. So the revolution of BTA tool center is the dominantfactor to generate the lobe profile of the hole. The dominantmode appears in higher amplitude of its frequency mode. So,
ig. 4. Predicted lobe profile of one cycle for half frequency withfa = 0.5 (Ra/Rw =
the most important factor that governs the roundness lobecontours are the amplitude of its mode and the frequency ofspindle error motion,fa, or the linear workpiece vibration.All these results can be seen inFig. 5. By changing thecondition ofRa,Rw, fa andfw, many lobes of different modeshape are generated. But their intrinsic shapes have theirnatural modes in their motion with frequency offa and fw,the motion results in explicitly look-alike hole roundnesslobe profiles similar to experimental lobe profiles. To verifythe roundness properties of the predicted profile of the holeshown inFig. 5a–u, the proposed harmonic model consider-ing the spindle error and workpiece motion must determinethe harmonic roundness characteristics of these lobe profiles.As shown inFig. 5, a multi-lobe produces several lobes bythe combinational revolution of the BTA tool and workpiecemotion and at last merged into multi-lobe profiles. Evenif, both their clear frequency exists, the amplitudes ofRaandRw are also a dominant factors that decide the mergedlobe characteristics. A condition of larger amplitude in BTAvibration of tool and workpiece results in the dominant lobeshape and its results are shown inFig. 5. But in general, theworkpiece vibration is smaller than the vibration of spindleerror.
Fig. 6 shows the predicted roundness of different fre-quency condition ranging 3–26 intermittently. Also all othern rizedi tiono nessbE n orw twom ofile.T atedb Thish s fi-n Alltt howq ged.F e in-t udet heF
3.3. Amplitude and frequency effect on roundness
Figs. 7–11shows the roundness predicted in several condi-tions of vibration amplitude and frequency by the suggestedmodel ranging 2–5. As the revolving radius of the spindletool error or the amplitude of workpiece vibration increase,the lobe profile generate more clear shape than low values ofRa andRw (Figs. 7–11). This phenomenon appears similarlyin other lobe more than six but it was not presented in thispaper. InFigs. 7–11, the higher amplitude of its roundnesswas generated in each case that its dominant amplitude andfrequency is involved in lobe profile. InFig. 10, the amplitudeof spindle error of the BTA tool is larger than that of work-piece vibration. Then, the dominant mode is the frequency ofspindle error motion.
3.4. Roundness mode detection by FFT
In Fig. 8, the amplitude of spindle error of the BTA toolis larger than workpiece vibration about five times, but thedominant frequency is the frequency of workpiece vibration.So the roundness profile does not show clearly.
In Fig. 11, the dominant amplitude and frequency are alsothose of workpiece vibration and it shows the same ratio ofroundness profile inFig. 11. Also it becomes clearer profilethan that ofFig. 7or Fig. 8. This phenomenon appears simi-l itsd pesi itialp
illedbr r ine pe ofh dt owerm
temt mea-s wellw i-ns
umber of modes can be calculated but it was not summan this paper. The spindle error motion and linear vibraf the workpiece can cause a various shape of roundy the combination of two different frequenciesfa and fw.ven if there are two clear modes in spindle error motioorkpiece, there is a tendency that does not to show itsodes by the phenomenon of merged effect in lobe prhe reason is that the final profile of the hole is genery the harmonic combination of two modes merged.armonic combination and merging of these two modeally generate the final roundness profile of the hole.
hese phenomena can be seen inFig. 6. It is also foundhat the combination of two modes have a tendency to suite a different number of modes by the modes merig. 6e shows in some sense a four-lobe mode but th
rinsic frequency having four-lobe does not really inclhe frequencyfa andfw. This phenomenon is similar in tigs. 3, 4 and 6.
arly in Figs. 5 and 6. But the global shape is decided byominant mode similarly in both conditions. All the sha
n above figure were obtained under the condition of inhase angle.φa andφw be set to zero for this case.
Fig. 12shows the natural mode of roundness in hole dry BTA tool calculated by FFT transform forFig. 6. Thisesult predicts well the exact lobe mode and its poweach case. It can be seen in figure that the whole shaole exactly match to that ofFig. 6. Using the FFT metho
he dominant roundness can be predicted well and its patch well with its dominant frequency.If this method is applied in roundness measuring sys
he above characteristics show good consistency withured one exactly well. All these output precisely matchith the lobes ofFig. 6. There is some difficulty in discrimating the number of lobe precisely inFig. 6, but it is easy toee the number of lobe exactly well by this method inFig. 12.
Fig. 13 shows the effects of the initial phase on lobeshape in case of three-lobe generation. The results meanthat the initial phases,φa and φw, of the BTA tool spin-dle error motion and workpiece vibration is dominant fac-tor on the lobe position but the whole shape has a sim-
Fo5(
ilarity in its shape having its intrinsic natural mode ofroundness.
3.6. Error estimation of the lobe
The radial error deviation in each lobe can be calculatedand it was represented inFig. 14. As the revolving frequency
ig. 5. Predicted lobe profile of the BTA hole ranging 1 to 6 with equal revolr 2, (c)fa/fw = 1/3 or 3, (d)fa/fw = 1/4 or 4, (e)fa/fw = 1/5 or 5, (f)fa/fw = 1/6 or 6/2, (k) fa/fw = 2/6 or 6/2, (l)fa/fw = 3/3, (m)fa/fw = 3/4 or 4/3, (n)fa/fw = 3/5 or 5/s) fa/fw = 5/5, (t) fa/fw = 5/6 or 6/5, (u)fa/fw = 6/6.
ute deviation with proposed model (Ra/Rw = 3/3). (a)fa/fw = 1/1, (b)fa/fw = 1/2, (g)fa/fw = 2/2, (h)fa/fw = 2/3 or 3/2, (i)fa/fw = 2/4 or 4/2, (j)fa/fw = 2/5 or3, (o)fa/fw = 3/6 or 6/3, (p)fa/fw = 4/4, (q)fa/fw = 4/5, (r) fa/fw = 4/6 or 6/4,
of the BTA tool center in spindle error motion increase theradial error that is the difference between average radius ofthe lobe, the basic workpiece radial error is decreasing atsome range and it increases again. In range of first step, thevalue range of coordinatefa is 0–1, also the value offw isranging from 1 to 11 in this area. In second step of ranging1–2 of fa in frequency coordinate, all values offa equals 2andfw is ranging from 1 to 11 in this second step. The nextseven steps are also classified in the same method. In thisfigure the least radial error condition is acquired in each stepat the condition offw about three.
The radial error deviation in each lobe is calculated underthe condition that one offa andfw is fixed. Also the predictedradial error may be represented inFig. 15. As the revolutionfrequency of BTA tool center in spindle error motion or work-piece motion increases, the radial error that is the differencebetween average radius of the lobe and the radius of basiccircle (hole radius) is decreasing up to some range and thenincreasing again. This means that the optimal frequency fordecreasing the radial error is ranging atfa = 66 (the similarresults atfw = 66). This range is the least error condition inthese lobe generations. Also, the radial error according to theincrease offa becomes larger than to the increase offw.
4
ing.In turedb ter
of the BTA tool used in experiments is 35 mm and the tipis P20 type. The experiments were performed at several cut-ting speed (V = 50, 60, 70 and 80 m/min) and feedrate (f =0.02, 0.03, 0.04, 0.05 and 0.06 mm/rev). After making thehole for roundness measurements, the roundness measuringinstruments was used for measuring the profile of the holeand its results were compared with the predicted one. As aresult, the predicted model is consistent well in lobe shapewith the measured roundness inFig. 18. With this compari-son, it was proved in our study that the profile of BTA drilledhole is the results of two main factors that are spindle er-ror motion of the BTA tool and linear motion of workpiecegiven. The multi-lobe profile is caused by the combination ofthese two factors. So, it can be concluded that the suggestedmodel for predicting the roundness is consistent well withexperimental measured lobe. And it can simulate all the realmulti-lobe that match with experimental one by the appropri-ate selection of factors such asRa, Rw, fa, andfw. In experi-mental results, two- to six-lobe profiles are most frequentlyobserved inFig. 18a–e. In BTA solid drilling, ten-lobe profileof different one had been observed as inFig. 18f. Also, theseresults show that the BTA drilling by feeding the tool in axialdirection includes many different shape of lobe in arbitraryposition according to its stiffness between workpiece and toolat that position.
obeo estedm tod anto um-b the
. Comparison of measured and predicted lobe
Fig. 16is the experimental set-up of the deep hole drillt was used to make the drilled workpiece ofFig. 17for round-ess test. This deep hole drilling machine was manufacy Shinil Machinery Co. (STGB-1500 CNC). The diame
In our analysis, it was concluded that every multi-lf several number was to be generated using this suggodeling. In some observation of lobe it is difficultiscriminate the number of lobe, which is more dominr weak. But the exact dominant and weak mode ner in the lobe of hole is clearly verified by adapting
FFT analysis. Also, the predicted lobe profile was com-pared with the real measured ones ranging from two- tosix-lobe that are most frequently observed in experimentsin Fig. 18. In experiments, the roundness tester (Rank Tay-lor Hobson Co.) was used for measuring BTA drilled holeprofile.
5. Conclusions
The roundness modeling of deep hole drilling was sug-gested in NC BTA drilling and its results are analyzed anddiscussed. The results of this paper are as follows:
1. A more realistic roundness lobe profile model has beendeveloped for predicting the resultant profile with respectto the error motion of the BTA tool spindle. By comparingthe predicted profile of the roundness model with the mea-sured one, both profiles are good consistency with eachother. So the proposed model is also a realistic model forroundness profile and the center of BTA tool spindle ro-tates in a type of harmonic contour motion in a stablerevolving state.
2. The amplitude of BTA tool radial error may decide thelargest and smallest boundaries of the round shape of thelobe. If there is a workpiece vibration in some frequency, it
ber of
ude,ss in
3 fre-g
ole.y beyolobe
4. The suggested harmonic roundness model may be appliedto any roundness shape machining such as turning, cylin-drical grinding or round shape machining. This roundnessperformance of the suggested model matches well with ex-perimental one and shows its reliability of the BTA drilledroundness model.
Acknowledgements
This work was partially supported by the Brain Korea 21Project in 2004.
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