Influence of Torsional Motion on the Axial Vibrations of a Drilling Tool

Sergey A. Voronov e-mail: voronov@rk5.bmstu.ru Alexander M. Gouskov e-mail: gouskov@rk5.bmstu.ru Alexey S. Kvashnin e-mail: a-kvashnin@yandex.ru Bauman Moscow State Technical University, 5, 2-nd Baumanskaya, Moscow, Russia 105005 Eric A. Butcher University of Alaska, P.O. Box 757500, Fairbanks, AK 99775 e-mail: ffeab@uaf.edu S. C. Sinha Fellow ASME Auburn University, 202 Ross Hall, Auburn, AL 36849 e-mail: ssinha@eng.auburn.edu Influence of Torsional Motion on the Axial Vibrations of a Drilling Tool The nonlinear dynamics of a tool commonly employed in deep hole drilling is analyzed. The tool is modeled as a two-degree of freedom system that vibrates in the axial and torsional directions as a result of the cutting process. The mechanical model of cutting forces is a nonlinear function of cutting tool displacement including state variables with time delay. The equations of new surface formation are constructed as a specific set. These equations naturally include the regeneration effect of oscillations while cutting, and it is possible to analyze continuous and intermittent cutting as stationary and nonstationary processes, respectively. The influence of the axial and torsional dynamics of the tool on chip formation is considered. The Poincaré maps of state variables for various sets of operating conditions are presented. The obtained results allow the prediction of conditions for stable continuous cutting and unstable regions. The time domain simulation allows determination of the chip shape most suitable for certain workpiece material and tool geometry. It is also shown that disregarding tool torsional vibrations may significantly change the chip formation process. DOI: 10.1115/1.2389212 1 Introduction Deep hole machining is one of the most complex manufacturing processes. Adverse conditions of chip formation, problems of chip removal, low stiffness of tool and its special design, and the impossibility of observing the tool during machining make the deep hole drilling as one of the most difficult operations. The main feature for holes of small diameter machining is the difficulty of removal of chips from the cutting region. Naturally, the chip removal becomes much easier in case of small crushed chip. A few methods of chip fragmentation have been recently discussed 1,2. The vibratory drilling is most efficient among them. This technology presumes installation of a special vibrator that excites the drill bit in its support fixture. The axial oscillations of cutting edges form a discontinuous chip if the correct amplitude and frequency of the vibrator are chosen. The new method of an autoresonant cutting technique has been developed for turning 3 and drilling 4, which implies that the vibration of tool by the self-exited vibration mechanism is inherent to the cutting processes. This method requires a more accurate analysis of tool dynamics and chip formation. This research has been carried out in Russia in the past several years and its features and model analyses are presented in several papers 3,5 8. The main attention was devoted to the modeling of axial tool vibrations as they define the chip formation process. But the tool is loaded by a torque as well as by a thrust force and thus can produce torsional oscillations too, which, under certain conditions, may have sufficient influence on the tool dynamics. The analysis of axial-torsional vibrations of the tool for deep hole drilling is considered in this paper. Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 17, 2006; final manuscript received September 1, 2006. Review conducted by Bala Balachandran. Paper presented at the ASME 2005 Design Engineering Technical Conferences and Computers and Information in Engineeromg Conference DETC2005, September 24 28, 2005, Long Beach, California, USA. The frequency analysis of drill vibrations has been presented in many papers. The bending vibrations of twisted beams under the action of an axial force and a moment were studied in 9. The influence of tool parameters on bending and torsional vibrations by the finite element method FEM was analyzed in 10. Rincon and Ulsoy 11 examined the influence of tool vibrations on cutting forces. But these authors did not include the interaction of cutting forces and tool vibrations in the model. Actually, the cutting forces arise due to the intrusion of tool edges into the material being cut and vary in proportion to chip thickness. Thus the cutting process dynamics is not a response to external loading, but to a self-excited process that can be stable or unstable. Gouskov et al. in 5,6 analyzed the stability of self-excited axial and transverse vibrations applying delayed differential equations. The stability lobes of drilling due to transverse 5 or axial drill bit vibrations 12 were determined by an asymptotic method. The research was extended by introducing equations of new surface formation 6,8 into the dynamical model that allow the simulation of nonstationary cutting processes, including discontinuous cutting when cutting edges disengage the machined material due to vibrations. It was shown that the bifurcation diagrams are useful in stability analysis and the prediction of dynamic behavior as a function of system parameters and the shape of the chip as well. The synchronization of tool vibration at the frequency of the external vibrator excitation in the case of vibratory drilling was analyzed in 7. Other researchers have demonstrated that in twisted beams the axial-torsional coupling exists and it can sufficiently influence the process of chip formation 13,14. As the drill vibrates in torsion, it lengthens and shortens periodically, causing a wavy surface formation on the bottom of the drilled hole. Bayly et al. 15 used a model that included torsional-axial coupling for frequency analysis to predict regions of instability. The simulation results were compared with experimental research and a close agreement was found. However, the illustrated model is suitable only for a stationary process of cutting with small amplitudes of tool vibrations when cutting edges are permanently in contact with workpiece 58 / Vol. 2, JANUARY 2007 Copyright 2007 by ASME Transactions of the ASME

edges into the workpiece material. The feed rate V and an angular speed are imparted to the workpiece. The full model of the system dynamics may be separated into three components: model of the tool dynamics; model of the new surface formation; model of the cutting forces. 2.1 Model of Tool Motion. The equations of tool motion in axial and torsional direction are obvious and do not differ from those presented in 15,17. These are given by Fig. 1 Photograph of the vibratory drilling setup: 1 vibrator; 2 tool; 3 workpiece material. If the vibration amplitudes are large, then cutting edges may lose contact with the material, introducing nonlinearity into the process 16. A similar model of coupled torsional-axial vibrations in 17 was presented, but the authors introduced a new cutting force model. The cutting force was considered distributed along the cutting edge and was determined as an empirical function of chip thickness at each point of the cutting edge. The chip thickness was determined by a numerical algorithm that disregarded the thickness variation due to tool torsional motion. This mechanical cutting force model was used in the time domain simulation of drilling with a pilot hole for an accurate prediction of vibrations that were compared against experimental results. This model can be used to study the dynamics of tool entry into and exit from the workpiece. Some other models of cutting processes are described in 18 20. The main difficulty, in tool simulation taking into account the coupled torsional-axial vibration, is that the time delay entering in the model of cutting force varies due to torsional vibrations. This fact requires using a special algorithm employing interpolation for chip thickness calculation, and it can sufficiently increase the calculation time. In the present paper a new algorithm is suggested by introducing the angle of tool rotation as a new independent variable, which is determined in time by an additional equation. In this case the integration by constant step size with constant delay makes calculations faster and more accurate. 2 Model Description The photograph of the vibratory drilling setup for machining holes in four connecting-rods simultaneously is shown in Fig. 1. A drill bit tool is considered a two-degree of freedom system that can vibrate in the axial with displacement w and torsional with twist angle directions. The schematic model of tool is presented in Fig. 2. The tool has a mass m and moment of inertia with respect to the z axis and has axial stiffness k ax and torsional stiffness k rot.an oscillatory motion imposed by an external vibrator as Z 0 t =à 0 sin 0 t, where à 0 and 0 are amplitude and frequency of the vibrator excitation, respectively, is imparted to the left support of the tool. The right end is under the action of cutting thrust force P C and torque M C arising during the penetration of tool cutting mẅ + d ax ẇ + k ax w + c w = mz 0 P C ; + d rot + k rot + c w = M C, 1 where w, are the generalized axial displacement and twisting angle; m, are tool mass and moment of inertia, d ax and d rot are generalized damping factors of axial and torsional vibrations correspondingly, and c w, c are coefficients characterizing axialtorsional coupling. The derivatives of state variables with respect to time t are denoted by dots. We assume that the tool of length l and workpiece are absolutely rigid, but the tool support is flexible. 2.2 Model of New Surface Formation. The tool has n C cutting edges arranged symmetrically at the right end of the tool. The nominal feed rate per one turn is determined as h 0 =VT, where T is the period of rotation T=2 /. The position of cutting edges in axial direction is determined as Z C t = Z 0 t + l + w t 2 The axial coordinate of surface L is measured in a coordinate system attached to the workpiece 8,16. For the following calculations it is more convenient to consider the coordinate of machined surface as a signal that comes to the cutting edge at time t, but it was formed by previous cutting edge J at time t T /n c when it cuts the same surface. For the jth-cutting edge at time t, this distance is equal to the surface coordinate of the cutting edge with number J= j 1 +n C fix 1/j, j=1,n C at time t T /n c, where fix 1/j denotes function that rounds the 1/ j to the nearest integer toward zero. It should be noted that T is not constant and depends on torsional vibrations and is equal to T only in the case when is equal zero or constant. Then the coordinate of the unmachined surface under the j-cutting edge is derived by the following relation: Z j t = Z 0 0 + l + H V t + L J t T /n c, where H is the distance from tool to the workpiece end mean surface at the initial instant and L J t T /n c is the depth of the hole surface under the Jth cutting edge measured from the workpiece left end. Let L j t, j=1,n C designate the depth of the machined hole along the generatrix under the jth cutting edge at the current instant. The surface L j t is formed due to cutting of instant chip thickness h j t of the surface that was formed by previous cutting edge J at time t T /n c : L j t = L J t T /n c + h j t. 4 In order to account for the possible disengagement of cutting edges out of material being cut, we introduce new variables D j t =Z C t Z j t as given by the distance from cutting edges to the unmachined surface 3 Fig. 2 Schematic model of a drill D j t = Z 0 t Z 0 0 + w t H + Vt L J t T /n C h j t =max 0,D j t 5 The second nonlinear relation in Eq. 5 defines the thickness of the undeformed chip. The initial function required to solve Eqs. 4 and 5 is given by Journal of Computational and Nonlinear Dynamics JANUARY 2007, Vol. 2 / 59

L j t = L 0J t, t 0. 6 In the case of an initially plane workpiece end that is normal to the hole axis, we should specify L 0j t =0, t 0. Obviously all cutting edges are in similar geometrical conditions. The full kinematics of vibratory drilling for any number of symmetrically disposed cutting edges and arbitrary law of tool axial motion is described by the set of equations 4 6. Inthe case of an absolutely rigid tool without elastic support and with a given law of support excitation, this set is sufficient for a calculation of the machined surface. In order to simplify calculations, the dimensionless variables are introduced by scaling displacement to h 0 and time to the period of tool rotation =t/t, 0 T=2 p, where p is the number of tool vibrations imparted by a vibrator during one workpiece rotation. The following dimensionless variables and parameters are used: = t T, = w h 0, = 2, f rot = T 2 k rot, = d ax 2 kax m, p = 0T 2, Z 0 = Z 0 h 0, A 0 = Ã 0 h 0, f ax = T 2 k ax m, H = H h 0, d rot 2 ; = m f d ax ax C = 2 3 2 f ax c w ; C = T2 h 0 h 0 2 c Now the kinematical equations in dimensionless form are given as j = Z 0 Z 0 0 + H + J T /Tn C ; j =max 0, j ; j = J T /Tn C + j ; j = 0j, 0; J = j 1 + n C fix 1/j. 7 2.3 Mechanical Model of Cutting Forces. There are several mechanical models of orthogonal and oblique cutting in the literature 1,12,15 17. In general, the cutting force is taken to be proportional to the uncut chip cross-sectional area. In case of drilling, the geometry of the cutting edge angles and speed are varied along the edge and the usual model of oblique cutting is not accurate. Altintas et al. in 17 used the distributed mechanistic model with empirical coefficients determined from a database using the orthogonal to oblique cutting transformation. This was applied to modeling of drilling with a pilot hole of diameter larger than the chisel edge. However, in this case the conditions along the edge do not vary significantly. In the case of usual drilling, the model used in 16 seems to be more efficient, as well as simpler. It satisfies similarity and dimensions theory and considers tool geometry with two empirical coefficients. Let us represent the axial component of the cutting force as follows see Fig. 2 : P C = k C0 h 0 /q q, 8 where k C 0 =g L Rq h 0 R/n C q 1 is the static axial cutting stiffness, g is a geometrical constant that is approximately unity g 1; L is the characteristic stress of workpiece material; R is drill radius; q is a nonlinear parameter 1 ; and is the reduced uncut chip thickness calculated by the following relation: Fig. 3 Stability diagram in parameters versus f ax = 1 n C n C q 1/q n C j. 9 j=1 Since the vibrator excitation is given by Z 0 =A 0 sin 2 p, the equations of tool motion for the chosen dynamic model disregarding disengagement of cutting edges out of cut material can be shown to be here derivatives by dimensionless time are dotted : +2 2 f ax + 2 f ax 2 + c = 2 p 2 A 0 sin 2 p 2 f ax 2 q ; q + 2 2 f ax + 2 f rot 2 + c = P 2 f 2 ax q, q 10 where c, c are dimensionless coefficients of axial-torsional coupling and =k 0 C /k is dimensionless cutting stiffness. The resultant torque of cutting forces is considered to be proportional to the thrust by a dimensionless coefficient P. The damping factor of torsional vibrations is taken as the damping factor of axial vibrations multiplied by some factor. The full system model is given by a combination of the equations of system dynamics 10, and equations of new surface formation 7. The resulting system is a set of nonlinear differential equations with the varying state dependent delay entering in nonlinear algebraic equations. The analysis of such a system is rather complicated and can be performed only numerically. 3 Stability Analysis of Continuous Cutting The stability analysis of continuous cutting is carried out from the linearized equations of system 7 and 10 that reduce to a set of delayed first order differential equations DDE. It should be noted that for case of an uncoupled system s =0, and s =0 the stability analysis is completely defined by the analysis of the single DOF model. The analysis shows that the stability regions do not significantly differ from those obtained for a single degree of freedom system given in 7,16. This is explained by the fact that in stable cutting the twist angle is constant and does not change the conditions of chip formation. The domains of stability in parameters versus f ax in Fig. 3 are presented s =0, s =0. The torsional-axial coupling does not change the stability region if we vary coefficients c w and c. The only difference between 1-DOF and 2-DOF systems is the small change in eigenfrequencies of the coupled system 21. 4 Dynamical Equation of the Full Model For a numerical analysis of the system model, we transform Eq. 10 into a normal Cauchy form. The main difficulty in numerical 60 / Vol. 2, JANUARY 2007 Transactions of the ASME

integration of Eqs. 10 is the variation of time delay due to torsional vibrations, i.e., delay is state dependent. One option is to use a special algorithm of the delayed function approximation at each step of integration, which, unfortunately, also decreases the accuracy of the calculations and increases the time of integration. In order to avoid this problem, we introduced a new independent variable for integration by changing the time t to the full relative angle of tool and workpiece rotation that varies in time as: t = t+ t, and its derivative as: d /dt= +d /dt. Or in terms of dimensionless variables, we have =t/t, =2, =2, T =2, d /d =1+d /d. Further, we consider as the new independent variable. An additional differential equation for the time derivative takes the form = d d = 1 1+ d d =1 d d 11 Therefore Eqs. 10 transform to: x = A x 1 + Bb 12 where the derivative with respect to is denoted by, and the new state vector is represented as x = T and = 0 0 1 0 0 0 0 0 0 0 1 0 0 0 B 2 p c 2 f rot 2 0 2 2 f ax 0 0 0 0 0 1, 2 A 0 2 f ax 2 /q 0 P 2 f 2 ax /q 0 0 = A 2 f ax 2 c 2 2 f ax 0 0, b = sin 2 p q The equations of new surface formation now include the constant delay and are given as: j = Z 0 x 5 Z 0 x 5 0 + x 1 H + x 5 J 1/n C j =max 0, j j = J 1/n C + j j = 0j, 0; J = j 1 + n C fix 1/j 13 Equations 12 and 13 represent a set of delayed algebraic differential equations DADE with a constant delay. This system is integrated by the Euler method with a constant step and iterations. For numerical analysis the following typical values of system parameters were specified: n C =2, q=0.75, P =1.0, =1.0, H=0, c =c =0. For validation purposes, first, the simplified model of axial and torsional vibrations coupling was analyzed 22. It was assumed that there were imparted stationary torsional vibrations of the tool with dimensionless amplitude and frequency. In this case the second equation in 10 is dropped and new variable becomes a periodic function given by 22 =1/ 1+ cos 2 and the last equation of system 12 can be integrated independently. Thus we obtain the delayed dynamical system with parametric excitation. The delay is constant and it is independent of the period of the parametric excitation. If the amplitude of parametric excitation is equal to zero, we obtain tool rotation with a constant angular speed and system is purely self-vibratory that is unstable under certain values of system parameters Fig. 3. The frequency of tool self-vibrations is a little bit greater than the eigenfrequency of its axial vibrations. Obviously, we expect that the main parametric resonance should arise in the vicinity of 2f ax, regardless of the fact that our system is delayed and nonlinear. Actually, this phenomenon is also observed here to. The only difference is that in our system the self-excited vibrations are strongly nonlinear and, probably, we can regard the system to be synchronized at the frequency of torsional vibration excitation. The Poincaré map of tool axial vibration amplitudes and uncut chip thickness for varying amplitudes of excitation is shown in Fig. 4. The case when torsion excitation frequency is equal to the eigenfrequency of axial vibrations is considered. The results of the excitation amplitude modulation 0,0.8 at = f ax show that in case of constant tool rotation without torsional vibrations =0 we have periodic stationary axial vibrations with chip fragmentation. In case of 0.6 due to parametric excitation, we can have continuous cutting with varying chip thickness. Under given values of parameters the parametric excitation has the largest effect at 0.4 when the amplitude of axial vibrations increases and chip shape becomes irregular. These results confirm the suggestion that torsional motion should be considered for a more correct analysis of the problem. Fig. 4 Poincaré map of the amplitude of a axial vibrations; b chip thickness versus excitation amplitude Journal of Computational and Nonlinear Dynamics JANUARY 2007, Vol. 2 / 61

Fig. 6 Poincaré maps of uncut chip thickness versus tool axial stiffness: 1 single-dof system; 2 two-dof system Fig. 5 Poincaré maps of a axial vibration; b torsional vibration versus tool axial stiffness in case of self-exited vibration 5 Results From Numerical Simulation Since the full model is a complicated set of nonlinear equations with a time delay, the most satisfactory method for pictorial presentations of various system motions is through Poincaré maps by plotting the extreme points of the tool position or the reduced thickness of the uncut chip when steady-state vibration is reached. We can pick out the most interesting factors for technology design, such as amplitude of cutting edge vibration, cutting discontinuity, chip dimension, the process of chip segmentation regularity and its synchronization with the frequency of imposed external vibrations. In this paper, only the self-excited vibrations without any external excitation are analyzed, i.e., A 0 =0. The Poincaré map of the tool axial and torsional motion versus axial stiffness of tool is presented in Figs. 5 a and 5 b for the following typical values of the system parameters P =1, =1, =8.48 10 2, =0.06, f rot / f ax =7. These parameter values correspond to the point A of stability diagram Fig. 3 close to the stability lobe but a little bit higher * =7.74 10 2, i.e., we cross the stability lobes in the horizontal direction through point A. We can observe from Fig. 5 that the regions of stable continuous cutting and interrupted cutting are periodically repeated. The character of axial motion is close to mono harmonic and torsional motion is more complex. The Poincaré map of the chip thickness variation versus tool axial stiffness for two system models: 1 single DOF only axial motion is considered and 2 two DOF are compared in Fig. 6. As it is seen, the plots are similar. Both are with an interrupted chip, but the amplitude of chip thickness variation for single DOF system is lower. The variation of axial displacement and twisting angle versus angle of rotation,, is shown in Fig. 7. These plots are obtained for a system simulation with parameters corresponding to point A f ax =6.6 of the diagram in Fig. 3. It should be noted that for the considered case when f rot / f ax =7, the amplitude of torsional vibrations is small and does not essentially influence chip process formation. The Poincaré maps of the axial displacement and torsional vibrations versus the dimensionless cutting stiffness in Figs. 8 a and 8 b for the two-dof system are presented. The diagram corresponds to the cross section of the stability lobes Fig. 3 in the vertical direction through the point A. The following values of parameters were taken P =1, =1, f ax =6.6, =0.06, f rot / f ax =7. The Poincaré map of the chip thickness variation versus the dimensionless cutting stiffness for two system models: 1 single DOF and 2 two DOF are compared in Fig. 9. We can see here that for the small magnitude of cutting stiffness, continuous cutting is stable, and at point * =7.74 10 2, we have a bifurcation point corresponding to an unstable region of the linearized system and the system vibration occurs with amplitude that is Fig. 7 Variation of a axial displacement and b twisting angle versus angle of rotation 62 / Vol. 2, JANUARY 2007 Transactions of the ASME

Fig. 8 Poincaré map of a amplitude of axial displacement; b amplitude of torsional vibration; versus dimensionless cutting stiffness sufficient for discontinuous cutting. If we compare results of the single-dof system and two-dof system, it is observed that we have a lower amplitude of chip thickness for the second case and amplitude increases if we increase the cutting stiffness. As a practicing engineer, one would be interested in how the ratio of torsional to axial stiffness f rot / f ax affects tool vibrations. The Poincaré maps of the tool axial and torsional motion versus f rot / f ax are presented in Figs. 10 a and 10 b for the following values of the system parameters: P =1, =1, f ax =2.7, =8.48 10 2, * =7.74 10 2, =0.02. The Poincaré map of the chip thickness versus f rot / f ax is shown in Fig. 11. We can see that the largest effect of torsional motion occurs when the ratio f rot / f ax is an integer and the influence is greater if the ratio f rot / f ax is lower than 5. In case when the ratio is greater than 5, the tool motion is regular at all ranges of the ratio f rot / f ax variation, except the values when f rot / f ax are close to an integer 5 or6or7, where the axial and torsional vibrations become polyharmonic and the chip Fig. 10 Influence of torsional stiffness on a amplitude of axial displacement; b amplitude of torsional vibration; f ax =2.7, =0.02, n=6000 rpm becomes irregular. At the same time, if we further increase f rot / f ax, then the region of irregular motion vanishes. In the case when the ratio is lower than 5, the tool motion is more complicated. The torsional vibration amplitude sufficiently increases and the amplitude of axial vibrations decreases and for some intervals 3.2, 3.4, 3.5, 3.6, 4.6, 4.8 becomes close to zero. The energy at these points transfers to the torsional vibrations. But the uncut chip amplitude here is small and cutting is intermittent. The result obtained confirms that in the case of autoresonant vibratory drilling when we apply the special tool holder with an additional elastic element we increase the magnitude of the ratio f rot / f ax and thus we make the process more regular with a lower amplitude of torsional vibrations. Usually the high frequency torsional oscillations decrease tool life, and so the autoresonant vibratory drilling becomes more efficient with a suitable choice of system parameters. It should be pointed out that the suggested algorithm does not work at lower values of the ratio f rot / f ax because an increase in torsional vibration amplitude leads to the cases where the absolute angular velocity of the tool becomes negative and the nu- Fig. 9 Poincaré map of chip thickness amplitude versus dimensionless cutting stiffness, : 1 single-dof system; 2 two-dof system Fig. 11 Influence of torsional stiffness on chip thickness amplitude f ax =2.7, =0.02, n=6000 rpm Journal of Computational and Nonlinear Dynamics JANUARY 2007, Vol. 2 / 63

merical integration procedure stops as time stops at the constant value. Actually in this case cutting edges are moving in an opposite direction to the tool rotation along the surface that was formed by the same tool edge some time before. Cutting conditions are significantly changed and time delay is not constant in spite of applying the new independent variable. The results obtained here show that the cutting force models disregarding influence of variable delay, used by Bayly et al. 15 and Roukema, Altintas 17 are incorrect in cases when the ratio of torsional to axial frequencies is lower than 5. 6 Conclusions A new model of coupled torsional-axial oscillations of vibratory drilling is presented. The model considers continuous and discontinuous cutting as well as the influence of new surface formation equations. The model permits the simulation of nonstationary processes and the prediction of cutting forces, displacements, and chip shapes for a specified set of system parameters. The introduction of an absolute angle of rotation as the new independent variable, instead of time, increases the system dimension, but leads to a set of equations with constant delay. Therefore, the integration is more accurate and effective. The stability analysis of the 2-DOF system shows that the torsional vibrations do not significantly change the stability lobes of continuous cutting, but can considerably change the character of system vibrations and the shape of chips in the instability region. The same conclusion but for the two-dof model of milling process by Insperger et al. 23 was drawn. The numerical simulation of self-excited vibrations of the 2-DOF systems shows that when torsional to axial frequency ratios are greater than 5, then the character of tool vibration in the region of unstable continuous cutting with respect to torsional vibration is similar to the same case in a 1-DOF system. The region of regular periodic motion decreases and the amplitude of chip thickness is lower. For all cases where f rot / f ax 5 the torsional motion plays an important role, as it essentially affects the process of chip formation. Especially, it should be considered in the case when tool axial stiffness is close to an integer multiple of torsional stiffness. Due to internal resonance, the energy of external excitation redistributes between axial and torsional modes and the chip formation process character changes. The amplitude of the uncut chip decreases and the motion of the system become highly irregular. Acknowledgment This research is financially supported by the National Science Foundation under Grants No. CMS-0114500 and No. CMS- 0114571. References 1 Stephenson, D. A., and Agapiou, J. S., 1997, Metal Cutting Theory and Practice, Marcel Decker, New York. 2 Poduraev, V. N., 1970, Cutting with Vibrations, Mashinostroenie, Moscow, p. 352. 3 Poduraev, V. N., and Kibalchenko, A. V., 1993, The Technology of Defense Industry for Manufacturing of Customer Goods, Rosconversia, Moscow, p. 528. 4 Brun-Picard, D., and Gouskov, A., Tête de Perçage à Effets Vibratoires, Patent B036, INPG. 5 Gouskov, A. M., Svetlitsky, V. A., and Voronov, S. A., 1979, Transverse Self-Vibration Excitation of Tool Used For Deep Hole Drilling, Collection of Papers Raschety Na Prochnost, Mashinostroenie, Moscow, 20, pp. 172 182. 6 Gouskov, A. M., Voronov, S. A., and Nikitin, A. S., 1992, Stochastic Regimes in Technologic Cutting Processes, Proc. 2nd International Scientific Technical Conference Actual Problems of Fundamental Science, Technosphera Inform, Moscow, BMSTU, 2, pp. B2 B5. 7 Gouskov, A. M., Voronov, S. A., and Batzer, S. A., 2000, Chatter Synchronization in Vibratory Drilling, Dynamics, Acoustics and Simulations/ASME 2000, 68, pp. 263 270. 8 Batzer, S. A., Gouskov, A. M., and Voronov, S. A., 2001, Modeling Vibratory Drilling Dynamics, ASME J. Vibr. 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A., Kvashnin, A. S., Butcher, E. A., and Sinha, S. C., 2005, Influence of Torsional Motion on Vibratory Drilling, Proc. IDETC/CIE 2005, 24 28 September, Long Beach, California, US. 22 Gouskov, A. M., Voronov, S. A., Batzer, S. A., and Sinha, S. C., 2002, Nonlinear Oscillations of a Tool Used in Deep Hole Drilling, Proc. 4th EURO- MECH Nonlinear Oscillations Conference, 19 23 August, Moscow, Russia. 23 Insperger, T., Stepan, G., Hartung, F., and Turi, J., 2005, State Dependent Regenerative Delay in Milling Processes, Proc. IDETC/CIE 2005, 24 28 September, Long Beach, California, US. 64 / Vol. 2, JANUARY 2007 Transactions of the ASME