One serious problem in deep-hole drilling is the occurrence of a dynamic disturbances called spiralling. A common explanation for the occurrence of spiralling is the coincidence of time varying bending eigenfrequencies of the tool with multiples of the spindle rotation frequency. We propose a statistical model for the estimation of the eigenfrequencies derived from a physical model. the major advantage of the statistical model is that it allows to estimate the parameters of the physical model directly from data measured during the process. This represents an efficient way of detecting situations in which spiralling is likelyy and of deriving countermeasures

Raabe, Nils

Webber, Oliver

Weihs, Claus

Eldorado - Ressourcen aus und für Lehre, Studium und Forschung

Deriving a statistial model for the preditionof spiralling in BTA deep-hole-drilling from aphysial modelC. Weihs1, N. Raabe1, and O. Webber21Chair of Computational Statistis,University of Dortmund, Germany2Department of Mahining Tehnology,University of Dortmund, GermanyAbstrat. One serious problem in deep-hole drilling is the ourrene of a dy-nami disturbanes alled spiralling. A ommon explanation for the ourrene ofspiralling is the oinidene of time varying bending eigenfrequenies of the toolwith multiples of the spindle rotation frequeny. We propose a statistial model forthe estimation of the eigenfrequenies derived from a physial model. The majoradvantage of the statistial model is that it allows to estimate the parameters ofthe physial model diretly from data measured during the proess. This representsan eÆient way of deteting situations in whih spiralling is likely and of derivingountermeasures.1 IntrodutionDeep hole drilling methods are used for produing holes with a high lengthto diameter ratio, good surfae nish and straightness. For drilling holes witha diameter of 20mm and above, the BTA deep hole mahining priniple isusually employed (VDI (1974)). The neessarily slender tools, onsisting of aboring bar and head, have low dynami stiness properties. Therefore deep-hole-drilling proesses are at a high risk of dynami disturbanes suh asspiralling, whih auses a multi-lobe-shaped deviation of the ross setion ofthe hole from absolute roundness, see g. 1.As the deep hole drilling proess is often applied during the last produ-tion phases of expensive workpiees, proess reliability is of prime importane.Predition and prevention of spiralling are therefore highly desirable.By using a nite elements model to determine drilling depth dependentbending eigenfrequenies of the tool, spiralling was shown to reproduiblyour when one of its slowly varying eigenfrequenies intersets with an un-even multiple of the tool rotational frequeny (Gessesse et al. (1994)). Thissuggests preventing spiralling by avoiding these ritial situations. Unfortu-nately the pratial appliation of the nite elements model is limited as ithas to be alibrated using experimentally determined eigenfrequenies.Earlier investigations demonstrated that the ourses of the bending eigen-frequenies learly show in spetrograms of the struture borne sound of the2 Weihs et al.Fig. 1. Left: Longitudinal setion of a bore hole showing marks resulting fromspiralling. Right: Assoiated roundness harts.boring bar, whih an be reorded during the proess (Raabe et al. (2004)).In this paper this signal is used to statistially estimate the parameters ofa physial model of the bending eigenfrequenies. A lumped mass model isused to alulate the tools bending eigenfrequenies. It inludes the physialparameters of the proess allowing to diretly alulate the inuenes of theirvariations on the eigenfrequeny ourses. However, this model ontains someunknown parameters and naturally the measurement is subjet to random er-ror. It is therefore ombined with a statistial model allowing the estimationof the unknown parameters by the Maximum Likelihood method.The work presented in this paper is based on experiments arried out ona CNC deep hole drilling mahine type Giana GGB560 (see Szepannek et al.(2006) for tehnial details). Self exited torsional vibrations were preventedthrough the appliation of a Lanhester-damper. The damper was moved atfeed speed together with the boring bar, implying a onstant axial positionof the damper relative to the tool. In order to detet bending vibrationsourring during the proess, time series of the lateral aeleration of theboring bar were reorded. The experiments were arried out with stationarytool and rotating workpiee. The experimental setup is illustrated in g. 2.Fig. 2. Experimental setup (top) and proposed modelling approah (bottom).Deriving a statistial model from a physial model 32 Physial ModelFor formulating the model the BTA system was redued to its most importantomponents. These are the tool, the Lanhester damper, two oil seal ringswithin the oil supply devie and the workpiee, again see g. 2, top. Underoperating onditions, the latter omponents at as lateral elasti onstraintsof the boring bar. While the damper stays in the same loation relative tothe boring bar, the oil supply devie is kept at onstant distane relative tothe workpiee and therefore moves at feed speed relative to the boring barduring the proess. The workpiee permanently ats on the tip of the tool.Fig. 3. Detailed modeling priniple: Regular linear elasti hain with additionallinear elasti support.As illustrated in g. 3 by an exemplary system with 4 degrees of freedom,the bar is subdivided into N elements of idential length l for onstrutingthe lumped mass model. These elements are linked to form a regular linearelasti hain omprising N identially spaed and elastially linked masses.Additional linear elasti supports represent the onstraints resulting from thesupporting elements. Adopting the x-oordinates as generalized oordinatesand assuming only small deetions we an write the homogenous equationsof motion of the system as[M ℄fxg+ [K(lB)℄fxg = f0g with [K(lB)℄ = [KTool℄ + [KSupp(lB)℄;where [M ℄NNand [K(lB)℄NNare the mass and stiness matries of thesystem and lBrepresents the atual drilling depth. The stiness-matrix anbe deomposed into the stiness matrix [KTool℄ of the boring bar and a ma-trix [KSupp(lB)℄ ontaining the stiness inuenes of the supporting elements.[KTool℄ is time onstant and an be omputed from the physial and geomet-rial properties of the tool (see Szepannek et al. (2006)), whereas [KSupp(lB)℄hanges stepwise with inreasing drilling depth due to the movement of theoil supply devie relative to the boring bar. Furthermore, [KSupp(lB)℄ gener-ally is unknown. More preisely, all elements of [KSupp(lB)℄ are zero exept ofthese elements on the main diagonal that orrespond to a supporting element4 Weihs et al.in the setup. The values of these matrix entries are the unknown parametersof the model.The stiness inuenes of the workpiee and the two seals of the stu-ing box within the oil supply devie are eah assumed to at pointwise andare therefore modelled by one single parameter eah (kwp, ksbf1;2g). TheLanhester-damper ontats the boring bar within a region of nominal lengthld. It is assumed, that this region may be redued, e. g. by wear. So two pa-rameters Æld;rand Æld;lrepresenting a right- and left-hand trunation of ldare added. The stiness inuene of the damper (kd) is equally distributedover the elements within the remaining region of length ld  Æld;r  Æld;l.The stiness onstants kwp, ksbf1;2g, kdtogether with Æld;r, Æld;l, whihdene the matrix [KSupp(lB)℄, are a priori unknown and annot be measureddiretly. These parameters therefore have to be estimated. For alulatingthe eigenfrequenies from the model the homogeneous equations of motionof the system (see above) have to be solved for eah regarded value of thedrilling depth lB. This leads to the following eigenvalue-problem [K(lB)℄  !2[M ℄fxgei!t:The solution of this problem onsists of the eigenvalues !2r;lB, the N squaredeigenfrequenies of the model, and the eigenvetors f	gr;lB, the orrespond-ing N mode shape vetors.3 Statistial ModelFor the estimation of the unknown parameters a statistial approah usingthe already introdued struture borne sound is proposed. In the followingthe data measured in a loation between damper and oil supply devie isexemplarily used.To provide a basis for statistial estimation of the unknown parametersp, the following statistial model is proposedSk(!; lB; p) = jjk(!; lB; p)j2j(!)2 S(lB):For eah value of the hole depth lBthe term Sk(!; lB; p) presents the peri-odogram of the struture borne sound measured at a loation orrespondingto element k. Due to the disreteness of the physial model lBhanges step-wise and so the periodograms are omputed based on non-overlapping time-windows. The model writes these periodograms Sk(!; lB; p) as the produtof a systemati omponent jjk(!; lB; p)j2j(!)2(the spetral density ofthe proess) and a stohasti exiting omponent S(lB), the periodogram ofa white noise proess. The systemati omponent onsists of the frequenyresponse funtion (FRF) series jk(!; lB; p) and the time onstant j(!),whih transforms the white noise proess into the exitation in element j.In a rst attempt j(!) for eah frequeny ! is set to its mean observedDeriving a statistial model from a physial model 5amplitude value. Renements like tting j(!) and p alternately are imag-inable in later investigations. For a better impression g. 4 gives a graphialrepresentation of the proposed statistial model.Sk(!; lB; p) = jajk(!; lB; p)j2aj(!)  2 S(lB)Fig. 4. Visualization of the statistial model.3.1 FRF ComputationFor the omputation of a FRF damping has to be inluded. The most straight-forward way of doing this is assuming proportional damping, implying thedamping matrix [C(lB)℄ = [K(lB)℄ + [M ℄. This leads to the two furthermodel parameters  and . Therefore the list of model parameters readsp = (kwp; ksbf1;2g; kd; Æld;r; Æld;l; ; ):Computation of the FRF neessitates the denition of the points of exitationj and response k. The exitation point j was hosen to be the last element N ,beause at this position the utting proess takes plae. Element k naturallyorresponds to the point at whih the onsidered signal is reorded. The FRFan then be omputed byjk(!; lB; p) = !2NXr=1	jr;lB	kr;lBkrr;lB  !2mrr+ i!rr;lB;where 	jr;lBdenotes the j-th element of the r-th mode shape vetor f	gr;lB,and krr, mrrand rrare the r-th diagonal elements of the modal stiness-, mass- and damping matries, respetively. These matries an diretly bederived from quadrati forms of the mode shape vetors and the stiness- andmass matries [M ℄ and [K(lB)℄ (Ewins (2000)). Finally, the eigenfrequenies!0r;lBof the proportionally damped system are given by!0r;lB= !r;lBq1  (!r;lB=2 + = [2!r;lB℄)2:As for this desription one a spei p is hosen, the orresponding eigen-frequenies an be determined.6 Weihs et al.3.2 Maximum Likelihood EstimationThe parameters of the systemati model part an be estimated using theMaximum Likelihood method. The Likelihood-funtion an be derived byonneting the following well known results.1. The periodogram Ix() of eah stationary proess Xtwith a moving-average representationXt=1Xu= 1ut uwith1Xu= 1(1 + juj)juj <1implying the spetral density fx() =Puuei2u2has an exponentialdistribution at eah Fourier frequeny  with parameter 1=fx(). Thenperiodogram ordinates at dierent Fourier frequenies are asympotiallyindependent (Shlittgen and Streitberg (1999), p. 364).2. Eah stationary proess Xtwith ontinuous and for all  non-negativespetral density fx() has an innite moving-average representation(Shlittgen and Streitberg (1999), p. 184).Assumption 2 an be seen as fullled, as all inspeted spetrograms learlyshow values dierent from zero for all frequenies and time points. Assump-tion 2 substantially implies assumption 1, so for eah Fourier frequeny !and hole depth lBthe distribution funtion of S(!; lB; p) is approximativelygiven byd:f:(s) = f(!; lB; p) 1ef(!;lB;p) 1s;where f(!; lB; p) = jjk(!; lB; p)j2j(!)2.Using the asymptotial independene of periodogram ordinates of dif-ferent frequenies and assuming independene for dierent hole depths, theLog-Likelihood-funtion is given byLL(p) =XlBX!ln1f(!; lB; p) S(!; lB; p)f(!; lB; p)The ML-estimators are the set pMLof parameters maximizing this funtion.With these parameters the estimated eigenfrequenies an be derived as il-lustrated in the last two setions.The introdued model has been suessfully tted to dierent experimentsby using the searh-based method by Nelder and Mead (1965) for the maxi-mization of the Log-Likelihood-funtion. Fig. 5 shows an exemplary ompar-ison between an aeleration spetrogram and the bending eigenfrequeniesomputed from the tted model for a proess without spiralling.Even though the seond and third eigenfrequeny seem to over-estimatethe area of elevated amplitudes the pattern in the spetrogram is learlyrepresented by the t. So apart from possible model renements, these results,Deriving a statistial model from a physial model 7Fig. 5. Comparison between aeleration spetrogram and tted eigenfrequenies.whih are similar for all other experiments investigated up to now, supportthe onnetion of the physial model with the statistial model as a basis forestimating its parameters from spetrogram data.4 Summary and OutlookThe presented paper shows that a onnetion of a physial and a statistialmodel helps to estimate the bending eigenfrequenies of a deep-hole-drillingtool from data available during the proess. Bending eigenfrequenies areknown to ause spiralling when rossing multiples of the spindle rotationalfrequeny. By supervising the estimated eigenfrequenies, shifts in the proessdynamis an be deteted and ruial situations an be predited.The supervision may be possible within a bath prodution, where aftereah ompletely drilled workpiee the eigenfrequenies are heked and theneessity of parameter hanges is deided. In the atual form tting the modelis too time extensive to allow online intervention. But if the model an besimplied implying faster tting proedures, strategies suh as ontrol hartsfor the eigenfrequenies ould be feasible as well.Simpliations of the physial model are possible by onentrating on therelevant regions of the spetra or modiations of the disretization. Thestatistial model may be simplied by estimating the spetra for frequeny8 Weihs et al.bands instead of Fourier frequenies using onsistent estimates as introduedin Shlittgen and Streitberg (1999). Furthermore for a more eÆient way ofestimating the eigenfrequenies historial data may be used in onnetionwith the physial model.In future experiments roundness errors of the drilled workpiees will bemeasured at dierent equally spaed hole depth points. These measurementsrepresent a quantization of the eet of spiralling over time and so help toinvestigate the development of spiralling more losely in dierent situations.Here main features of interest are whether spiralling starts rapidly or devel-ops slowly and if the magnitude of spiralling depends on how quikly thefrequeny rossing takes plae. As the measurement of roundness errors isnot possible in prodution the potentials of estimating these from the spe-trogram data will be heked as well.AknowledgmentThis work has been supported by the Collaborative Researh Center 'Redu-tion of Complexity in Multivariate Data Strutures' (SFB 475) of the GermanResearh Foundation (DFG).ReferenesEWINS, D. (2000): Modal Testing: Theory, Pratie and Appliation., 2. ed., Re-searh studies press, Badlok.GESSESSE, Y.B., LATINOVIC, V.N., and OSMAN, M.O.M. (1994): On the Prob-lem of Spiralling in BTA Deep-Hole Mahining. Transation of the ASME,Journal of Engineering for Industry, 116, 161{165.NELDER, J.A. and MEAD, R. (1965): A Simplex Method for Funtional Mini-mization. Computer Journal, 7:308{313.RAABE, N., THEIS, W., and WEBBER, O. (2004): Spiralling in BTA Deep-HoleDrilling - How to Model Varying Frequenies. Conferene CD of the FourthAnnual Meeting of ENBIS 2004, Copenhagen.SCHLITTGEN, R. and STREITBERG, B.H. (1999): Zeitreihenanalyse. Olden-bourg, Munhen.SZEPANNEK, G., RAABE, N., WEBBER, O., and WEIHS, C. (2006): Preditionof Spiralling in BTA Deep-Hole Drilling - Estimating the System'S Eigenfre-quenies. Tehnial Report, SFB 475, Dortmund.VDI-Rihtlinie 3210 (1974): Tiefbohrverfahren. VDI Dusseldorf.

Deriving a statistical model for the prediction of spiralling in BTA deep hole drilling from a physical model

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Deriving a statistical model for the prediction of spiralling in BTA deep hole drilling from a physical model

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