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DOI 10.1007/s00170-003-2039-6 ORIGINAL ARTICLE Int J Adv Manuf Technol (2004) Jean-Luc Battaglia Andjrez Kusiak Estimation of heat fl uxes during high-speeddrilling Received: 23 March 2003 / Accepted: 27 May 2003 / Published online: 8 December 2004 Springer-Verlag London Limited 2004 Abstract Heat fl uxes on each cutting edge of a carbide double cutting drill are estimated during a high-speed machining pro- cess from temperature measurements in the drill tool and a direct model that has been established using the non integer system identifi cation approach. A single experiment is required in order to characterize the transient thermal behavior of the tool. The non integer system identifi cation method is based on the recur- sive linear least square algorithm. The inverse method is based on the constant function specifi cation approach. Results obtained during machining lead to predict the tool wear and possible tool positioning defect. Keywords Drilling process Inverse thermal problem Multivariable system Non-integer system identifi cation 1 Introduction The aim of this study concerns the estimation of the heat fl ux on each cutting edge of a drilling tool during high-speed machining as represented on Fig. 1. The machining parameters are the feed rate f and the cutting velocity Vc. Temperature and heat fl ux estimation in a tool during ma- chining has been essentially developed in the turning process framework. The fi rst set of techniques, presented in the works of Her- bert 1, Shore 2 and Stephenson 3 for example, consists of using the natural thermocouple formed by the workpiece and the tool when the two materials are electric conductors. Neverthe- less, as shown for example by Laraqi 10, a thermal resistance occurs at the sliding interface of the two materials. Thereby, the tool-workpiece thermocouple method does not lead to the J.-L. Battaglia (u) A. Kusiak Laboratoire Energ etique et Ph enomnes de Transfert, Ecole Nationale Sup erieure dArts et M etiers, UMR 8508, Esplanade des Arts et M etiers, 33405 Talence cedex, France E-mail: jlblept-ensam.u-bordeaux.fr Tel.: (+33)-05-56-84-54-21 Fax: (+33)-05-56-84-54-01 average temperature on the tool but on the average tempera- ture at the sliding interface. Jaspers et al. 4, Changeux 5, and Kwon et al. 6 use infrared (IR) thermography in an orth- ogonal cutting process confi guration. This measure is strongly dependant on the spatial variation of the emissivity coeffi cient on the aimed surface and consequently one cannot predict ac- curately the temperature magnitude. Nevertheless, it gives very interesting qualitative information concerning heat transfer in the cutting domain. The approach, presented by Stephenson 7, uses IR thermography and thermocouples together in order to calibrate the emissivity according to the temperature from the thermocouples. The second set of methods is based on the resolution of the inverse heat conduction problem in the tool. This approach con- sists of estimating the heat fl ux in the tool from temperature measured at one or several points located close to the heated area. Furthermore, it uses the transient thermal behavior model of the tool that expresses the temperature at the sensors accord- ing to the heat fl ux in the tool. Stephenson 8 and Stephenson and Ali 9 measure the temperature in the workpiece by IR thermography and use the analytical solution of heat transfer in the workpiece to estimate the temperature in the shearing zone. Groover and Kane 11 embed two thermocouples in the tool holder. A model that expresses the average temperature on the cutting edge and the temperature at these two thermocouples is achieved by a nodal method based on the thermoelectric anal- ogy. An insulating element is placed between the insert and the tool holder. Then, one considers that the insert is thermally insu- lated from the tool holder that led to limit the diffusion domain. A comparable methodology is proposed by Yen and Wright 12. In their application, the temperature at the thermocouple is ex- pressed according to the average temperature on the cutting edge from an exponential law.Parameters in this relation are identifi ed from temperature measurements on a specifi c apparatus that per- mit simulating variations of the temperature on the cutting edge. A comparable approach is found in the work of El-Wardany 13. There are few works in the literature concerning assess- ment of the heat fl ux in the tool during a drilling process, and more generally in all processes with a tool being in rotation. Fig.1. Schematic representa- tion of the drilling process; cutting parameters are the feed rate f and the cutting speed Vc Kim et al. 14 arrange a thermocouple in the workpiece and record the temperature when the fl ank face of the tool passes above the thermocouple. Lin 15 uses an indirect method lead- ing to the temperature of the workpiece in the shearing zone during a milling process. Its method is based on the tempera- ture measurement at a point of the machined surface behind the tool using an IR pyrometer and a model of heat transfer in the workpiece. The method presented in this paper consists of estimating the heat fl ux on eachcutting edge of a drill rotating from temperature measurements in the tool. The particularity of our approach is that the direct model that expresses the temperature at the sensors according to the heat fl uxes has been established from the non integer system identifi cation approach. Such an approach has already been exploited to estimate heat fl uxes applied on each insert of a milling tool during machining (see 16 and 17). Section 2 presents the complete experimental device that has been developed for temperature measurement in the tool during a drilling process. In Sect. 3, the direct model describing heat Fig.2. Schematic representation of the experimental device: two thermocouples type K are embedded in the drill, nearest to the cutting edge of each cutting edge; each signal is amplifi ed from a AD595CQ chip and then return to the four slip rings rotating collector; the amplifi ed signals are sent to a NI acquisition card (PCMCIA, 12 bits) transfer in the tool is built using the non integer system identifi - cation method. The mathematical signifi cance of the non integer differentiation operator versus heat conduction in the systems is presented in the papers of Battaglia et al. 1820 and does not need to be recalled here. Section 4 presents the mathematical procedure for the system identifi cation stage. The experimen- tal device that has been developed for the system identifi cation stage is described in Sect. 5. The inversion procedure, in terms of the mathematical algorithm and of its robustness, is developed in Sect. 6. Finally, Sect. 7 is devoted to the application that es- timates the heat fl ux in a carbide double cutting drill during a high-speed machining process. 2 Experimentaldevice description Temperature measurement in a rotating tool imposes the use of a specifi c apparatus in order to ensure the link between each sensor and the acquisition device. This apparatus must not disturb the measure by introducing heavy noise and thermal drift. Concerning temperature measurement from a thermocou- ple, Stephenson 3 and Bourouga et al. 21 use rings with a mercury bath whose implementation requires a great amount of caution. On the other hand, the measure in the real machin- ing confi guration is strongly perturbed by vibrations given to the weak sensitivity of the thermocouple that is about some V per C. Micro-thermistors of micro series type have a greater sensi- tivity, about some mV per C. Unfortunately, these sensors can only be used from 50 to 150 C temperature range. Our solution consists of embedding two thermocouples inthe drill, which is equipped with two oil holes generally used to lu- bricate the tool during machining. These holes are used to locate the thermocouples as close as possible to each cutting edge in order to improve the sensitivity of the temperature at the meas- urement locations according to the variations of the heat fl uxes. Obviously, only the duration of the heat fl ux variations greater than the sampling period t will be observed. The sensors are Fig.3. Photography of the experimental device fi xed with an adhesive epoxy monocomponent, containing sil- ver type Loctite 3880, which is characterized by its high thermal conductivity. In order to minimize the noise infl uence, electric signals that come from the thermocouples are fi rst amplifi ed from a specifi c device, based on the AD595CQ chip, which turns with the tool. The amplifi ed signals are transmitted toward the acquisition device by using high-speed split rings with four gold circuits. This rotating collector has been especially developed for this application and can be used for rotational speeds up to 3000 rpm. The complete experimental apparatus is represented in Figs. 2 and 3. The Labview software drives the acquisition de- vice, which is a PCMCIA National Instruments, 16 bits, 6061E type. 3 Design of the heat transfer model using the non integer system identifi cationmethod The classical resolution of the mathematical relations describing transient heat transfer in the drill requires knowledge of the ther- mophysical properties of the materials that constitute the drill. This led to the realization that there were as many experiments as unknown properties. Furthermore, heat transfer in the cutting domain is perturbed by the presence of sensors that are located very close to the heated surfaces of the tool. On the other hand, the sliding contact on the slip rings of the rotating collector can also signifi cantly perturb the measured signals. These consider- ations led to the consideration of the sensors behaviors in the heat transfer model of the tool, and the infl uence of the rotating collector with respect to the signal transmission. In the face of such diffi culties, the system identifi cation ap- proach appears to be more adapted and leads to more reliable results. It consists of identifying the parameters of a model that expresses the heat fl uxes applied on the tool according to the temperature at the sensors. The use of such an approach requires realizing a specifi c experiment that permits applying a measur- able heat fl ux on the tool area that is solicited during machining. The system can be identifi ed from a single experiment but, in practice, a second experiment is required in order to validate the model. Nevertheless, the model obtained from this approach is much more reliable that that obtained from several charac- terization experiments. On the other hand, the identifi ed system integrates the sensors infl uence and that of the rotating collec- tor. This means that the system identifi cation approach leads to the characterization of the heat transfer in the tool as well as the thermal experimental device. Furthermore, given that the same sensors are used during the heat fl uxes estimation procedure, there is no uncertainty on the sensors locations. Using the Laplace transform, the model that expresses the heat fl uxes according to the temperature at the sensors can be represented in the following form (see also Fig. 4): ?T 1(s) T1(s) ? = ?F 1,1(s)F1,2(s) F2,1(s)F2,2(s) ? ? F(s) ? 1(s) 2(s) ? .(1) According to previous results obtained in the fi eld of system identifi cation concerned with the diffusion process 18,19, it has been demonstrated that each transfer function Fi,j(s), in the transfer matrix F(s), is of the fractional form: Fi,j(s) = Lij ? k=Lj 0 ij k s k Mij ? k=Mij 0 ij k s k ,ij M0 = 1 ,(2) where = 1 2 .(3) The inverse Laplace transforms applied in Eq. 2 lead to the con- tinuous time expression of Ti(s) = Fi,j(s)j(s) as: Mij ? k=Mij 0 ij k D kTi(t) = Lij ? k=Lj 0 ij k D kj(t) .(4) In this relation, Df(t) = df(t) dt , with R, denotes the frac- tional derivative of function f(t) with respect to variable t. This operator can be view as the generalization of the classical deriva- tive of integer order. The reader can fi nd in references 22 Fig.4. Schematic representa- tion of the tool in terms of the inter infl uence graph; accord- ing to the precision on the thermocoupleslocationsin the tool, the functions B1(s) and B2 (s) can signifi cantly differ and 23 the basic mathematical defi nitions and properties in the fi eld of fractional calculus. The summation bounds ?Mij 0 , Mij, Lij 0 , Lij?, in relation Eq. 4, essentially depend on the location of the sensor i from the heated surface j. 4 Identifi cation procedure Let us denote ei(t) as the measurement error for sensor i at time t, defi ned by: yi(t) = Ti(t)+ei(t) .(5) Substituting the value of Ti(t) from Eq. 5 into Eq. 4 gives: D M ij 0 yi(t) = Lij ? k=Lj 0 ij k D kj(t) Mij ? k=Mij 0 +1 ij k D kyi(t)+i(t) .(6) It appears that the residue i(t) is expressed from the non integer derivatives of the measurement errors as: i(t) = Mij ? k=Mij 0 ij k D kei(t) .(7) Equation 6 can be represented in the form of a linear regression as: D M ij 0 yi(t) = H(t) +i(t) ,(8) with H(t) = ? D ? Mij 0 +1 ? yi(t)DM ij yi(t)D L ij 0j(t) D L ij j(t) ? ,(9) and = ? ij M0+1 ij M ij L0 ij L ?T .(10) If one considers K +1 successive measures, where t is the sampling period, then Eq. 8 becomes D M ij 0 YK= HK+EK,(11) with YK= yi(t) yi(t+t) yi(t+ K t) ,HK= H(t) H(t+t) H(t+ K t) andEK= (t) (t +t) (t + K t) .(12) Thus, the estimation of the unknown vector , that minimizes the scalar quantity J = ?EK?2, is obtained in the linear least square sense as: = ? HT KHK ?1 HT KD Mij 0 YK.(13) In practice, the regression vector is not built from the successive derivatives of the temperature measurement, given that they am- plify the measurement error. A more effi cient approach consists of using the fractional integration of the data. This representation is easily obtained by considering = 1 2 instead of 1 2 in Eq. 4. In practice, K is high, generally some thousand, and it is bet- ter to use the recursive form of Eq. 13 that consists of expressing the identifi ed parameters at time t from those estimated at the previous time (t 1), see 24. The sequential algorithm is: (t) =(t1)+L(t) ? DM ij 0 yi(t)H(t) (t 1) ? ,(14) with L(t) = P(t1) H(t)T (t)+H(t) P(t1) H(t)T ,(15) and P(t) = P(t1) P(t1) H(t)TH(t)P(t 1) (t)+H(t)P(t 1) H(t)T .(16) The initial values are (0) = 0Nand P(0) = 106IN, where 0N and INare respectively the nil vector and the identity matrix whose dimensions are N = Mij Mij 0 +LijLij 0 +1. The value of (t) is fi xed at one, which signifi es that all the samples have the same weight. The Kalman theory shows that the covariance matrix of the identifi ed parameters is: cov ? ? = P ?t f ? y 2 ,(17) where tf denotes the fi nal time and the standard deviation of the measurements is approximated by: y EKET K K .(18) Fig.5. Experimental device for the system identifi cation of the drilling tool: as represented on the zoom image, the heat resistor is fi xed on one heated area whose dimensions have been measured after the machining operation; the heat fl ux in the tool is assumed to be equal to the electric power provided to the heat resistor; a switch permits generating a pseudo-random signal in order to improve the reliability of the identifi ed system at the small times as well as the long times (duration of the process) 5 Experimental device for the system identifi cation The identifi cation of the drilling tool is achieved from a specifi c apparatus, represented on Fig. 5, which allows for controlling and measuring the heat fl ux applied on one cutting edge. The sur- face is heated by a micro resistance (5.7 ) formed by a platinum circuit embedded on an alumina plate of 250 m thick. The con- tact between the insert and the micro-resistor is realized using a silver charge stick type Loctite 3880. As it is represented in Fig. 5, the small dimensions of the micro-resistor and the stick are designed to heat the surface corresponding to the tool-chip sliding contact area. On the other hand, the thermal inertia of the heating apparatus does not exceed 0.1s. In this experimental procedure, the slip ring has no rotation and we assume that the identifi ed system remains unchanged during machining. This assumption has been verifi ed by meas- uring the electric resistance at each slip ring with and without rotation. On the other hand, the tool is fi xed during the system iden- tifi cation experiment, which does not exactly reproduce the real boundary conditions on the tool during machining. In fact, the thermal exchange coeffi cient on the tool does not vary signifi - cantly with the rotation, but thermal radiation can occur between the tool and the workpiece that can lead to an increase in the tem- perature at the sensor. Nevertheless, the machining duration is less than 5 s and the temperature at the sensors is not sensitive to a change on the boundary condition at the surface of the tool that does not contribute to the cutting process. Finally, it must be noted that the temperature reached on the heated surface during the identifi cation stage is generally less than that obtained during machining. Thereby, the linearity as- sumption cannot be verifi ed. 6 Heat fl uxes estimation The estimation method consists of assuming that the unknown vector ?(t) = 1(t)2(t)T(19) is constant from time t to time (t +rt), where t is the sam- pling period and