Generalized modeling of drilling vibrations. Part II: Chatter stability in frequency domain

Transcription

1 International Journal of Machine Tools & Manufacture 7 (7) Generalized modeling of drilling vibrations. Part II: Chatter stability in frequency domain Jochem C. Roukema a Yusuf Altintas b a Department of Mechanical Engineering Manufacturing Automation Laboratory The University of British Columbia 65 Applied Science Lane Vancouver BC Canada V6T1Z b Department of Mechanical Engineering Manufacturing Automation Laboratory The University of British Columbia 65 Applied Science Lane Vancouver BC Canada V6T1Z Received 18 September 6; accepted October 6 Available online November 6 Abstract A time domain model of the drilling process and hole formation mechanism is presented in Part I and the general solution of drilling chatter stability in frequency domain is presented in this paper. The drill s flexibility in torsional axial and lateral directions are considered in determining the regenerative chip thickness. Stability is modelled as a fourth order eigenvalue problem with a regenerative delay term. The critical radial depth of cut and spindle speed are analytically determined from the eigenvalues of the characteristics equation of the dynamic drilling process in frequency domain. The method is compared against the extensive numerical solutions in time domain which were presented in Part I cutting experiments and previously published partial stability laws. The time domain model presented in Part I of the paper considers tool geometry dependent mechanics all vibration directions and the true kinematics of drilling while allowing for nonlinearities such as tool jumping out of cut and nonlinear cutting force models. It is shown that accurate prediction of drilling stability requires modeling of drill/hole surface contact stiffness and damping which is still a research challenge. r 6 Elsevier Ltd. All rights reserved. Keywords: Chatter; Stability; Frequency domain; Torsional; Axial; Lateral 1. Introduction The drill is a slender pretwisted beam clamped by a tool holder in the spindle and in contact with the metal that is being cut. The lateral axial and torsional vibrations of the drill cause an irregular distribution of the chip thickness which depends on the drill edge geometry as well as the present position and the past history of the cutting edge location which are governed by the rigid body motion and vibrations of the drill bit. An efficient frequency domain solution is advantageous for quick stability analysis of cutting operations. Tobias [1] Tlusty [] and Merit [] were the first to establish chatter stability laws for one dimensional cutting operations. These stability laws are Corresponding author. Tel.: ; fax: addresses: roukema@interchange.ubc.ca (J.C. Roukema) altintas@mech.ubc.ca (Y. Altintas). URL: applicable to operations like orthogonal turning where the chip thickness generation is time invariant. Their theories explained the mechanism of chip thickness regeneration as the main source of unstable chatter vibrations. The chatter vibration free depth of cut was shown to be inversely proportional to the cutting stiffness and real part of the frequency response function between the tool and workpiece. For the more complex milling process where forces and the direction of excitation vary time domain solutions were developed by Tlusty et al. [5] and Montgomery et al. [6] which included nonlinearities such as the tool jumping out of cut due to excessive vibrations. However in order to understand the mechanism and quickly predict chatter free cutting conditions analytical models for milling have been developed by Minis et al. [7] and Budak et al. [8]. Drilling time domain simulation models have been developed by Arvajeh et al. [91] to simulate torsional axial and lateral chatter. Roukema et al. have presented extensive numerical simulation models of the drilling process as well [11 1] /$ - see front matter r 6 Elsevier Ltd. All rights reserved. doi:1.116/j.ijmachtools.6.1.6

2 J.C. Roukema Y. Altintas / International Journal of Machine Tools & Manufacture 7 (7) Nomenclature B dynamic drilling coefficient matrix (dimensionless) B first order approximation of dynamic drilling coefficient matrix (dimensionless) B red reduced drilling coefficient matrix (dimensionless) b radial depth of cut [mm] b lim critical depth of cut [mm] C damping matrix of drill bit [Ns/m] c xx c yy viscous damping [Ns/m] D drill diameter [mm] D p pilot hole diameter [mm] du tool deflection in the direction of the cutting lips [mm] dxdydzdy regenerative tool displacements [mm] F t F r F a tangential radial and axial force acting on flute [N] F u F v forces on drill bit in frame rotating with the tool [N] F x F y F z cutting forces acting on tool tip [N] f r feedrate [mm/rev] G p frequency response matrix in rotating frame [m/ N] h chip thickness [mm] h s static chip thickness [mm] dh 1 dh change in chip thickness on flute 1 [mm] K stiffness matrix of drill bit [N/m] K c cutting stiffness matrix frequency domain solution [N/mm ] k m number of modes included in a vibration direction (dimensionless) k h modal stiffness [N/m] k tc tangential cutting stiffness [N/m ] k rc k ac radial and axial force cutting stiffness factors (dimensionless) k xx k yy lateral stiffnesses [N/m] k ZF z direct axial stiffness axial deflection over axial force [N/m] k ZT c cross axial stiffness axial deflection over torque load [Nm/m] k yt c direct torsional stiffness torsional deflection over torque [N/rad] k yf z cross torsional stiffness torsional deflection due to thrust loading [N/rad] M mass matrix of drill bit [kg] m xx m yy lumped mass at drill tip [kg] N f number of teeth on drill (dimensionless) n spindle speed [rev/min] R drill radius [mm] R p pilot hole radius [mm] R t torque arm (torque from tangential and radial forces) [mm] Dr vector with regenerative displacements [mm] T tooth period [s] T c cutting torque acting on tool tip [Ncm] t time [s] x c y c z c lateral and axial tool tip deflections [mm] b xx b xy b yx b yy lateral coefficients of dynamic drilling coefficient matrix (dimensionless) b zz b zy b yz b yy torsional axial coefficients of dynamic drilling coefficient matrix (dimensionless) g g 1 g g coefficients characteristic equation [N/m] a torsional axial coupling factor (dimensionless) e phase shift between vibration marks of consecutive teeth [rad] z h modal damping ratio (dimensionless) y c torsional tool tip deflection [degrees] k ratio of imaginary part and real part of eigenvalue (dimensionless) LL R L I eigenvalue of characteristic equation real and imaginary parts [N/m] F(s) transfer function [N/m] F ab cross transfer function [N/m] F XX F YY direct lateral transfer functions [N/m] F ZZ F yy direct axial and torsional transfer functions [N/ m] F Zy F yz cross axial and cross torsional transfer functions [N/m] f st entry angle of the drill [rad] f ex exit angle of the drill [rad] f p pitch angle of the drill [rad] c phase shift of eigenvalue [rad] O angular speed of the tool [rad/s] o c chatter frequency [rad/s] natural frequency [rad/s] o nh Although the model by Roukema et al. [1] can simulate lateral chatter whirling vibrations and torsional axial chatter and visualize forces and the resulting hole bottom and wall finish it is computationally costly to predict chatter stability lobes. Compared to milling spindle speeds in drilling are low and natural frequencies in the tooltoolholder-spindle system are high thus requiring fine grids for numerical stability [1]. Hence a frequency domain stability law that takes all vibration modes encountered in drilling into account would allow for rapid prediction of chatter free cutting conditions. Galloway [15] analyzed chatter vibrations on radial drilling machines which have flexibility in the axial direction of the bit. He also provided a stability chart with experimental verification. Bayly et al. [16] developed chatter stability models for drilling holes with large pilot hole sizes and reaming operations which were restricted to bending vibrations of the tool. Their simulation model showed three- and five-sided holes resulting from lateral chatter vibrations in drilling. Bayly et al. [17] developed a frequency domain solution for

3 176 ARTICLE IN PRESS J.C. Roukema Y. Altintas / International Journal of Machine Tools & Manufacture 7 (7) torsional axial chatter in drilling by extending Tlusty s stability law for orthogonal cutting [18]. The torque acting on the drill unwinds the drill while the bit elongates at the same time which leads to a chip thickness regeneration mechanism in axial direction. Bayly et al. also developed models to predict the hole shape resulting from reaming operations [19] and drilling [1]. Dilley [] focused on lateral chatter in drilling. Arvajeh and Ismail [91] modeled bending stability in drilling by assuming the drill tip is pin supported in the hole being cut which was proposed by Ema et al. []. Tool rotation at the tip provided for a chip thickness regeneration mechanism that explained bending instability with vibration frequencies three to four times higher than the lateral bending frequency. This paper presents a global stability solution for dynamic drilling that takes all flexibilities into account. The effects of two lateral torsional and axial vibrations on the regenerative chip load are considered. The dynamics of the drilling process is formulated by four sets of coupled delayed differential equations. The critical radial depth of cut and spindle speeds are evaluated from the eigenvalues of the characteristic equation describing the stability of the drilling system. The proposed method which is compared against extensive numerical solutions and previously z c y c xc θ c Y deformed drill geometry Fig. 1. Dynamic model of drill bit drill bit deflections. k y Z X k z k x k θ published partial stability solutions is the first comprehensive model of the drilling stability in the literature.. Dynamic drill bit model The general equations of motion for the dynamic drilling system can be formulated in the stationary frame as follows: x c ðtþ _x c ðtþ x c ðtþ F x ðtþ >< y c ðtþ >= >< _y c ðtþ >= >< y c ðtþ >= >< F y ðtþ >= ½MŠ þ½cš þ½kš ¼ z c ðtþ _z c ðtþ z c ðtþ F z ðtþ >: y c ðtþ >; >: _y c ðtþ >; >: y c ðtþ >; >: T c ðtþ >; (1) where (x c y c ) denote the lateral (z c ) the axial deflections of the drill in the global coordinate system as illustrated in Fig. 1.(y c ) the torsional deflection of the drill bit itself with respect to the rigid body spindle motion. The rotation speed of the tool is O in [rad/s]. The matrices [M] [C] and [K] contain the lumped mass damping and stiffness characteristics reflected at the drill tip respectively (Fig. 1). The external loads acting on the drill include two lateral forces (F x F y ) thrust force (F z ) and torque (T c ). The dynamic properties for the drill used in this paper are provided in Table 1. While higher stiffness and damping values are used for the time domain simulations shown in Figs. 6 8 experimentally identified modal parameters are used in predicting torsional axial chatter stability presented in Fig. 9. As explained in the experimental section the real stiffness and damping values are significantly higher than those obtained from modal tests on a drill with free end. The drill end is in contact with the material during the drilling process which significantly alters the dynamic stiffness of the drill structure. The sign in the table indicates whether the tool deflection is positive or negative. Thrust compresses the drill whereas torque extends the drill by untwisting it. This sign is taken into account in the proposed frequency domain solution.. Dynamic chip thickness in three dimensional drilling The (positive) force components acting on each flute are tangential (t) radial (r) and axial (a) as illustrated in Fig. Table 1 Dynamic properties of drill bit Mode Frequency (Hz) Fig. 6 8 Fig. 9 versus experiments Sign Stiffness Unit Damping (%) Stiffness Unit Damping (%) XX [N/m]. N.A. N.A. N.A. N.A. YY [N/m]. N.A. N.A. N.A. N.A. k ZF z [N/m] [N/m].5 positive k yt c [Nm/rad] [Nm/rad].5 positive k ZT c [Nm/m] [Nm/m].5 negative k yf z [N/rad] [N/rad].5 negative

4 J.C. Roukema Y. Altintas / International Journal of Machine Tools & Manufacture 7 (7) F t Y b du spindle axis F r dh dh 1 F a X 1 h s h s1 κt Ω F a1 1 Ωt R r p Fig.. Elemental forces acting on the cutting edges of a two-fluted drill bit. and defined as F t1 ¼ k tc bh 1 ; F r1 ¼ k rc F t1 ; F a1 ¼ k ac F t1 F t ¼ k tc bh ; F r ¼ k rc F t ; F a ¼ k ac F t ðþ where h 1 is the uncut chip thickness for flute 1 measured in spindle axis direction b is the radial depth of cut defined by the difference of tool radius and pilot hole radius: b ¼ R R p. The radial and axial forces are expressed to be proportional to the tangential force. The total cutting forces acting in X Y and Z directions at the tool tip are (see Fig. ): F x ðtþ ¼ðF t1 F t Þ sin Ot ðf r1 F r Þ cos Ot F y ðtþ ¼ðF t1 F t Þ cos Ot ðf r1 F r Þ sin Ot F z ðtþ ¼F a1 þ F a T c ¼ R t ff t1 þ F t ðf r1 þ F r Þg where R t is the torque arm for calculating the cutting torque T c from tangential and radial forces. The dynamic chip thickness is influenced by vibrations in three orthogonal directions and one torsional direction. The static chip thickness equals the feed per revolution divided by the number of flutes which is two in this case (N f ¼ ): h s ¼ f r. () N f The change in chip thickness due to regenerative displacements dx dy on each flute are: dx cos Ot dy sin Ot dh 1 ¼ tan k t ðdx cos Ot dy sin OtÞ dh ¼ ð5þ tan k t where k t is the tip angle of the drill and Ot is the tool rotation angle. An increase in chip thickness on flute 1 is accompanied by an equal sized decrease in chip thickness on flute. The chip thickness change is illustrated in Fig. where du ¼ dx cos Ot dy sin Ot is the tool deflection in the direction of the cutting lips. F t1 F r1 ðþ Fig.. Chip thickness change due to lateral vibrations while drilling a piloted hole. The regenerative displacements are: dx x c ðtþ x c ðt TÞ >< dy >= >< y c ðtþ y c ðt TÞ >= fdrg ¼ ¼ (6) dz z c ðtþ z c ðt TÞ >: >; dy >: y c ðtþ y c ðt TÞ >; where T ¼ p/n f O is the tooth period. The dynamic chip thickness due to torsional vibrations (positive in the direction of tool rotation) is expressed for two teeth as follows: dh 1 ¼ dh ¼ dy 1 p f r (7) and depends on the feed per revolution f r. Finally axial vibrations influence the chip thickness directly: dh 1 ¼ dh ¼ dz. (8) The total change in chip thickness becomes ( ) dh 1 ¼ dh 8 1 tan k t ðdx cos Ot dy sin OtÞ dz f r < : 1 tan k t ðdx cos Ot dy sin OtÞ dz f r p dy p dy 9 = ;. ð9þ The static chip thickness h s is neglected since it does not contribute to the stability. The dynamic forces depend on the dynamic chip thicknesses dh 1 and dh : 8 9 >< >: F x F y F z T c 8 >= ¼ >; d k tc bðdh 1 dh Þ sin Ot k tc k rc ðdh 1 dh Þ cos Ot >< k tc bðdh 1 dh Þ cos Ot k tc k rc ðdh 1 dh Þ sin Ot >= k tc k ac bðdh 1 þ dh Þ >: fk tc bðdh 1 þ dh Þgð1 k rc ÞR >; t 9 ð1þ

5 178 ARTICLE IN PRESS J.C. Roukema Y. Altintas / International Journal of Machine Tools & Manufacture 7 (7) where the sum of dynamic chip thicknesses (dh 1 +dh ) and difference (dh 1 dh ) are: ( ) ( dh 1 þ dh dz f ) r p dy ¼. (11) dh 1 dh tan k t ðdx cos Ot dx sin OtÞ When the dynamic chip thickness is substituted in Eq. (1) the dynamic forces acting on the tool are found to be: F x dx >< F y >= >< dy >= ¼ bk F tc ½BðtÞŠ (1) z dz >: >; >: >; dy T c where the dynamic drilling coefficient matrix [B(t)] depends on time t spindle speed O cutting coefficients k rc and k ac tool tip angle k t torque arm R t and feed per revolution f r. Because the forces in X and Y directions depend on the chip thickness difference (dh 1 dh ) axial and torsional deflections do not affect the lateral cutting forces (F x F y ) d. Similarly as the torque and thrust depend on the chip thickness sum (dh 1 +dh ) lateral tool deflections do not affect the dynamic thrust force F z or torque T c. The dynamic cutting forces can be summarized as: ffðtþg ¼ bk tc ½BðtÞŠfDrg (1) where {Dr} contains the regenerative displacements from Eq. (6). The time dependent matrix [B(t)] is periodic at tooth passing frequency N f O or tooth period T ¼ p/n f O and can be expanded into Fourier series: by taking the mean value as was done for milling by Altintas et al. [8] (Fig. ) ½B Š¼ 1 T Z T ½BðtÞ dtš (1) [B ] is valid only between entry (f st ) and exit (f ex ) angles of the drill which are and p as a drill cuts throughout the drill pitch angle f p ¼ p/n f. b xx b xy ½B Š¼ 1 Z fex b yx b yy ½BðfÞ dfš ¼ f 6 p f st b zz b zy 7 5. (15) b yz b yy Directional factor [-] bxx bxy byx byy Cutter rotation angle [degrees] Fig.. Time varying directional coefficients for drilling. The integration only needs to be done for sub-matrix [B 11 ]: b xx ¼ N Z f 1 ðsin f þ k rc k rc cos fþ df p tan k t ¼ N f pk rc p tan k t b xy ¼ N Z f 1 p ð 1 þ cos Ot þ k rc sin OtÞ df p tan k t ¼ N f p p tan k t b yx ¼ N Z f 1 p ð1 þ cos Ot þ k rc sin OtÞ df p tan k t ¼ N f p p tan k t b yy ¼ N Z f 1 p ð sin Ot k rc þ k rc cos OtÞ df p tan k t ¼ N f pk rc. ð16þ p tan k t The coefficients of sub-matrix [B ] are rewritten as b zz ¼ N f p ð pk acþ b yz ¼ N f p f pð1 k acþr t g b zy ¼ N f p ð k acf r Þ b yy ¼ N f p f ð1 k acþf r R t g. ð17þ Factoring out term N f /p the directional matrix [B red ]is expressed as: pkrc p tan k t tan k t p pkrc ½B red Š¼ tan k t tan k t 6 pk ac k ac f 7 r 5 f pð1 k rc ÞR t g f f r ð1 k rc ÞR t g (18) and the dynamic drilling force equation becomes ffðtþg ¼ N f p bk tc½b red ŠfDrg. (19). Frequency domain solution procedure The frequency response function matrix F(io) at the tool tip is identified as F xx ðioþ F yy ðioþ FðioÞ ¼ 6 F zz ðioþ F zy ðioþ 7 5 () F yz ðioþ F yy ðioþ assuming there is no coupling between the lateral directions individually and no cross talk from lateral directions into axial and torsional directions either. F aa (io) is the direct

6 J.C. Roukema Y. Altintas / International Journal of Machine Tools & Manufacture 7 (7) frequency response function and F ab (io) is the cross frequency response function which is defined as follows: F ab ðioþ ¼ Da F b ¼ Xkm h¼1 o nh =k h o nh o þ iz h o nh o (1) where ab (a or b is xyz or y) denotes displacement of cutter tip coordinate a when a cutting force F b is applied in direction b. k m is the total number of structural modes in the system h represents each of these modes and o nh k h and z h are the natural frequency modal stiffness and damping ratio respectively. Following Budak s method [8] the drilling system is critically stable when the harmonic regenerative displacements {Dr} occur at the chatter frequency o c with a constant amplitude: fdrðio c tþg ¼ frðio c tþg frðio c t io c TÞg ¼ð1 e ioct Þ½Fðio c ÞŠfFge ioct ðþ where o c T is the regenerative phase delay between the vibrations at successive tooth periods T. Substituting {Dr(io c t)} into the dynamic drilling Eq. (19) gives: ffge ioct ¼ N f p bk tc½b red Šð1 e ioct Þ½Fðio c ÞŠfFge ioct () which has a nontrivial solution if the determinant is zero: det ½IŠ N f p bk tcð1 e ioct Þ½B red Š½Fðio c ÞŠ ¼. () This is the characteristic equation of the closed loop dynamic drilling system. By defining the oriented transfer function matrix as F ðio c Þ¼½B red Š½Fðio c ÞŠ b xx F xx b xy F yy b yx F xx b yy F yy ¼ 6 b zz F zz þ b zy F yz b zz F zy þ b zy F yy 7 5 b yz F zz þ b yy F yz b yz F zy þ b yy F yy ð5þ and the eigenvalue L of the characteristic equation as L ¼ L R þ il I ¼ N f p bk tcð1 e ioct Þ (6) the characteristic equation becomes detð½išþl½f ðio c ÞŠÞ ¼. (7) The eigenvalues of Eq. (7) can be found by scanning through possible chatter frequencies o c around the natural modes. The critically stable depth of cut (b lim ) and spindle speed n can be found by a similar procedure presented by Budak and Altintas [8]. b lim ¼ pl R N f k tc ð1 þ k Þ;! k ¼ L I L R T ¼ 1 ð þ kpþ;! ¼ p tan 1 k! n ¼ 6 o c N f T. ð8þ Because matrix (Eq. (5)) has decoupled lateral and torsional axial terms the characteristic equation (Eq. (7)) can be simplified to: " # " #! " #! 1 Lb xx F xx Lb xy F yy 1 þ 1 Lb yx F xx Lb yy F yy 1 " # Lðb zz F zz þ b zy F yz Þ Lðb zz F zy þ b zy F yy Þ þ Lðb yz F zz þ b yy F yz Þ Lðb yz F zy þ b yy F yy Þ ¼. The roots are then found by solving: ðlateralþðtorsional axialþ ¼ zfflfflfflfflfflfflfflfflfflfflfflfflffl} fflfflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ ðg L þ g 1 L þ 1Þ ðg L þ g L þ 1Þ ¼ g ¼ðb xx b yy F xx F yy b xy b yx F xx F yy Þ g 1 ¼ðb xx F xx þ b yy F yy Þ g ¼ðb zz b yz F zz F zy þ b zz b yy F zz F yy þ b zy b yz F yz F yz þ b zy b yy F yz F zy b yz b zz F zz F zy b yz b zy F zz F yy b yy b zz F yz F zy b yy b zy F yz F yy g ¼ðb zz F zz þ b zy F yz þ b yz F zy þ b yy F yy Þ ð9þ ðþ resulting in the following four roots: pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi L 1 ¼ g 1 þ g 1 g g ; L ¼ g 1 þ g 1 g g ðlateralþ; pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi L ¼ g þ g g g ; L ¼ g þ g g ðtorsional axialþ: g (1) 5. Comparison with alternative partial chatter stability laws Bayly et al. presented chatter stability laws for torsional axial vibrations and lateral vibrations separately in a series of noted articles [16 1]. Bayly et al. [16] modeled the lateral chatter stability of drilling using rotating coordinates. A modified version of that approach is compared against the proposed method presented in this paper while neglecting the torsional axial vibrations. As shown in the following the modeling of drill dynamics in the rotating frame eliminates the time variation of drilling coefficients hence the justification of taking average values in the proposed method can be verified Lateral chatter stability using a rotating coordinate system In a fixed coordinate system (XY) the equations of motion of the drill are expressed as " m xx #( ) x " c xx #( ) _x m yy y þ c yy _y " k xx #( ) x ( ) F x þ k yy y ¼ F y ðþ

7 18 ARTICLE IN PRESS J.C. Roukema Y. Altintas / International Journal of Machine Tools & Manufacture 7 (7) where proportional damping is assumed. A rotating frame of reference (UV) is attached to the tool where the U- direction is aligned with the cutting lips as illustrated in Fig. 5. The frame rotates with a constant spindle speed O [rad/s] (Fig. 5). The system dynamics can be converted into the rotating frame using: ( ) " #( ) ( ) x cos Ot sin Ot u F x ¼ ; y sin Ot cos Ot v F y " #( ) cos Ot sin Ot F u ¼ ðþ sin Ot cos Ot F v where the instantaneous rotation angle is y ¼ Ot. By substituting the displacement velocity acceleration and forces into Eq. () the drilling dynamics can be expressed in the rotating frame as " #( ) " #( ) m xx u c xx m xx O _u þ m yy v m yy O c yy _v þ k xx m xx O ( c xx O 5 u ) ( ) F u c yy O k yy m yy O ¼ ðþ v where the system has now time invariant but speed dependent matrixes. The variation of the cutting forces on the drill depends on the current vibration u(t) and the vibration u(t t) one tooth period earlier and is expressed in matrix form as ( ) F u F v " ¼ k #( ) tck rc b= tan k t uðtþþuðt tþ k tc b= tan k t vðtþþvðt tþ F v ( ) uðtþþuðt tþ ¼ b½k c Š vðtþþvðt tþ (5) where the current vibration and the vibration one tooth period earlier are summed as the displacements of opposing teeth have opposite signs in the rotating frame. It is assumed that vibrations in the tangential direction (V) do not affect the chip thickness. When the system is critically stable the stability of the drilling dominated by Ω Y Fig. 5. Definition of rotating and stationary coordinate systems. V Ωt U X the lateral vibrations is reduced to: I þ bg p ðo; OÞ½K c Šð1 þ e iot Þ ¼ (6) where G p (o;o) ¼ [ o M+ioC(O)+K(O)] 1 is the frequency response matrix of the system in the rotating frame: G p ¼ k xx m xx O m xx o þ oc xx i c xx O þ m xx Ooi c yy O m yy Ooi " ¼ G # uu G uv G vu G vv 5 k yy m yy O m yy o þ oc yy i. ð7þ The term (1+e iot ) in the characteristic equation arises instead of (1 e iot ) found in classical turning stability because in the rotating frame the displacements of opposing teeth have opposite signs. Although the equation for the chatter stability solution appears to be a matrix equation it is in fact a scalar equation due to the zero column in the K c matrix. For a given spindle speed O chatter frequencies o are found for which the imaginary part of Eq. (6) vanishes and the corresponding depth of cut is calculated from: tan k t b lim ¼ k tc RefG uu ðo; OÞk rc þ G uv ðo; OÞg. (8) The minimum value of b lim for each speed O is considered the critical radial depth of cut. Equation (8) resembles Tlusty s one dimensional chatter stability law [18] which leads to a positive real depth of cut and spindle speed as explained in [18]. Equation (8) is very similar to the lateral stability law proposed by Bayly et al. in [16] except that it does not depend on feedrate and is valid for any radial depth of cut. 6. Chatter stability lobes for drilling The chatter stability of drilling is evaluated by comparing the proposed method with time domain simulations previously published partial stability laws and experiments. The natural frequencies and damping ratios presented in Table 1 are used. The stiffness and damping for both lateral and torsional axial modes has been increased to provide a clear comparison between the proposed frequency domain solution and the previously developed time domain solution [1] for the 5 5 rpm speed range. The cutting coefficients were determined from drilling a 16 mm hole which has been predrilled with a mm drill bit. The feedrate is.15 mm/tooth. The tangential force F t and radial force F r were measured using a single fluted drill (one flute has been ground away). The thrust force F z and torque T c were measured using a regular drill with two flutes. F t ¼ 1111:7N; F r ¼ 1:7N; (9) F z ¼ 59:9N; T c ¼ 1:199Nm: The thrust torque and lateral forces are assumed to be linearly dependent on the radial depth of cut (b) to be used 1

8 J.C. Roukema Y. Altintas / International Journal of Machine Tools & Manufacture 7 (7) in the frequency domain solutions presented here. The cutting coefficients are provided in Table Comparison of proposed frequency domain solution with time domain model Fig. 6 shows the comparison between the proposed frequency domain solution and time domain simulation results at 5 cutting conditions from the numerical model developed by Roukema et al. [11]. The time domain model presented in Part I of the paper uses nonlinear cutting coefficients but is based on the same workpiece material and drill geometry [1]. The tool is given a three micrometer radial runout to create lateral force unbalance that can result in lateral chatter (otherwise the forces on the two flutes would cancel each other out and lateral deflections would not occur). From individual inspection of the lateral axial and torsional tool deflection histories the time domain results are classified as stable when tool Table Cutting force model parameters proposed frequency domain solution; V c ¼ 1 m/min f r ¼. mm/rev pilot ¼ mm Parameter Value k tc N/m k rc k ac R t.7 (dimensionless).9 (dimensionless).66 m deflections are transient/reach a steady state value and do not grow indefinitely. The classification lateral chatter is given when only lateral chatter occurs torsional axial chatter when only torsional axial chatter occurs and unstable when both lateral and torsional axial chatter develop at the same time [1]. The lateral chatter stability curve predicts a minimum critical depth of cut of.8 mm. In time domain however the process is still stable at. mm depth of cut over the whole speed range (which is clearly above the stability border). While the frequency domain solution is linear the time domain model considers the nonlinearities such as cutting edge and chip load dependent cutting force coefficients and tool jumping out of cut due to excessive vibrations. Hence the frequency domain solution gives more conservative stability estimate than the time domain model. The torsional axial stability curve predicts a minimum critical depth of cut of. mm which is more close to the time domain solution. At 11 rpm the drilling process is stable up to 6.5 mm depth of cut coinciding with stability pockets of both lateral and torsional axial chatter stability. At 5 1 and 1 rpm torsional axial chatter develops ahead of lateral chatter as these speeds are located at the lateral chatter stability pockets but above the torsional axial stability border. At 1 rpm lateral chatter develops from. mm depth of cut up to 6.5 mm and there is no torsional axial chatter because these conditions fall within the torsional axial stability pocket. At and rpm torsional axial and later chatter develop at the same time. The stability border b c a 5. Depth of cut [mm] torsional-axial stability border lateral stability border time domain simulation results: stable lateral chatter unstable torsional-axial chatter Spindle speed [krpm] Fig. 6. Comparison of proposed frequency domain with time domain simulation results.

9 18 ARTICLE IN PRESS J.C. Roukema Y. Altintas / International Journal of Machine Tools & Manufacture 7 (7) Cross-sectional profile at pilot hole Cross-sectional profile at pilot hole Cross-sectional profile at pilot hole Lateral [μm] Lateral [μm] 6. x1 x1 x f c =6Hz f s =167Hz f s =18Hz Axial [μm] Axial [μm] x1 x1 x f c =1Hz Torsional [deg] Torsional [deg] (a) 5. x1 x1 x f c =1Hz (b) (c) Fig. 7. Time domain simulation details for three cases in Fig. 6: (a) 1 rpm 5.5 mm depth of cut lateral chatter; (b) 1 rpm 5.5 mm depth of cut torsional-axial chatter; (c) 11 rpm 5.5 mm depth of cut stable cut.

10 J.C. Roukema Y. Altintas / International Journal of Machine Tools & Manufacture 7 (7) trends identified from time domain results clearly follows the frequency domain solution. Fig. 7 provides details of three time domain simulation cases denoted by a b and c in Fig. 6. Each column shows the three-dimensional workpiece finish the cross-section of the surface at the pilot hole lateral deflection and frequency spectrum and axial and torsional deflections with their spectra. Case a shows lateral chatter developing very quickly as can be seen by the lateral deflections and the side profile of the surface. The number of waves on the surface is small as the chatter frequency is about 6 Hz. Case b shows torsional axial chatter at about 1 Hz leaving about 19 waves on the surface. The lateral deflection stabilizes at about mm after full engagement due to the runout on the drill at spindle frequency f s ¼ 167 Hz [1]. Case c is stable and results in a smooth surface while small lateral deflections at f s ¼ 18 Hz occur again due to the drill runout. 6.. Proposed frequency domain model versus partial stability laws For clarity the stability solution of the proposed method is split into a torsional axial stability and a lateral stability part. The proposed torsional axial stability is compared with Bayly s work [17] in Fig. 8a whereas the lateral stability is compared with the rotating coordinate frame approach in Fig. 8b. In order to use Bayly s solution for torsional axial chatter [17] the torsional axial coupling factor a needs to be calculated. The static axial deflection for the drill bit (Table 1) due to thrust and torque would be 59:N 1:199 Nm dz ¼ 1: N=m :6 1 5 N=m ¼ :87 :5 mm ¼ 18:69 mm: ðþ Hence the torsional axial coupling factor the drill bit is determined as 18:69 mm a ¼ ¼ :87 (1) :87 mm which shows that axial extension due to torque overwhelms the compression due to thrust. The thrust cutting constant is determined as 59:N k c1 ¼ :6 :15m ¼ 565:8 16 N=m () a and k c1 are used to calculate the torsional axial chatter stability from [17] 1 b lim ¼ ak c1 Re½GðoÞŠ () using the natural frequency of f n ¼ 57.5 Hz damping ratio z ¼.5% and stiffness of [N/m] for transfer function G(o) in the axial direction. Bayly s torsional axial chatter stability law predicts a more conservative depth of cut. The lobe shape is slightly (a) 7 6 Depth of cut [mm] torsional-axial (Bayly) torsional-axial (proposed) Depth of cut [mm] 5 lateral (proposed) lateral (rotating coordinates) 1 (b) Spindle speed [krpm] Fig. 8. Comparison between proposed frequency domain solution and (a) Bayly s torsional-axial stability [17]; (b) lateral stability for stationary tool by Bayly [16] and lateral stability using rotating coordinates modified version of Bayly s approach [16].

11 18 ARTICLE IN PRESS J.C. Roukema Y. Altintas / International Journal of Machine Tools & Manufacture 7 (7) different from the proposed method but the lobe locations line up well. The shape of the lobes predicted by the proposed method are different as it takes four different transfer functions into account rather than lumping them into one. Readers are referred to Bayly s paper [17] for details. For the lateral stability chart comparison two solutions are shown. The lateral chatter stability predicted using a rotating coordinates approach (modified version of Bayly s approach [16]) closely matches the proposed solution (Eq. (8)). The rotating coordinates approach uses time invariant cutting coefficients while the proposed method takes the average of the time varying coefficients. They result in the same stability border which justifies the averaging applied in Eq. (16). 6.. Proposed frequency domain solution versus experimental results Experiments presented by Roukema [11] indicate that torsional axial chatter is always dominant and lateral chatter does not occur for the investigated drill which has a length/diameter ratio of 11. Fig. 9 compares the predicted torsional axial stability lobes (the proposed solution Eq. (8)) with the experimental results. The tool dynamics are provided in Table 1 (these are different tool dynamics than used for Figs. 6 8 and the speed range is also different here). Since it is difficult to measure the dynamic stiffness correctly with impact testing the stiffness of the drill is evaluated from the finite element model of the twisted drill and the natural frequency and damping ratios are identified from impact modal tests [1]. The predicted stability pockets are very small and narrowly packed and cannot be used to select vibration free cutting conditions in practice. Experiments indicate that the drilling a full hole is always stable when the spindle speed is below rpm. Also when the chisel edge is in contact i.e. largest radial depth of cut the process is even more stable which is contradictory to the linear stability laws. The physics behind having stable drilling at low speeds and larger radial depth of cut is explained as follows. The chisel edge contact significantly increases the stiffness of the drill and its scratching effect at the hole bottom surface drastically increases the damping. The authors conducted impact modal tests when the drill tip is free chisel edge is in contact and the drill lip is engaged with the hole. Significant increases are observed both in stiffness and damping. However they ar still different than the drill s dynamic properties during cutting since the speed of cutting edge alters the contact stiffness and damping. The increase in the stability with the enlarged radial depth of cut is attempted to be explained by the argued contact stiffness and damping but it s physics need to be modeled before making a clear conclusion. Experiments indicate that as the speed is decreased the process also becomes more stable which is contrary to the linear stability prediction laws. The discrepancy is attributed to the process damping at low speeds as presented by Altintas and Weck []. If chatter occurred at 18 rpm the number of waves between consecutive teeth would be 56 resulting in significant friction between the waves left on the surface and flank face of the drill lips which is called process damping. If the speed is lower the number of waves increases further and the process damping effect becomes even stronger see Roukema et al. [1]. Neither the process damping nor the contact stiffness and damping of 8 7 full hole chisel edge does not cut Depth of cut [mm] stable (experiments) torsional-axial chatter proposed frequency domain torsional-axial chatter lobes Spindle speed [rpm] Fig. 9. Comparison of experiments and proposed frequency domain solution for torsional axial chatter stability; Tool dynamics: Table 1; Workpiece material: AL75-T751; feedrate ¼. mm/rev.; Flood coolant was used during experiments; Depths of cut mm in the chart.correspond to pilot hole diameters mm (the drill diameter is 16 mm.

12 J.C. Roukema Y. Altintas / International Journal of Machine Tools & Manufacture 7 (7) the tools with machined surface have been modeled in the literature and deserves to be studied by tribology researchers. 7. Conclusions A novel method to calculate the chatter stability lobes for drilling is proposed and compared against experiments and extensive time domain simulations using the model developed by the authors which takes all nonlinearities into account. The new method considers lateral torsional and axial vibrations and shows an good match with time domain simulation model presented in Part I [1] and previously published solutions. The experiments show that torsional axial chatter is dominant and lateral chatter is not dominant in drilling. For drills with larger length to diameter ratios lateral chatter may become important. It should be noted that lateral chatter stability in drilling is very different from milling in the sense that during stable cutting the resultant lateral force in drilling is zero due to cancellation of bending forces by the symmetric drill lips. However torsional axial chatter does have a steady state torque and thrust in drilling and contributes strongly to the chatter. Although the presented drilling stability model is most comprehensive in comparison to the partial stability laws published in the literature it is not able to provide close agreements with the experimental observations except providing similar trends. The accuracy of the proposed and previously published drilling stability models would greatly improve if the process damping and drill-hole contact stiffness are accurately modeled which is the current research challenge in dynamics of metal cutting at low speeds. Acknowledgements This research is sponsored by NSERC and Pratt & Whitney Canada. Guehring and Sandvik companies provided the cutting tools and tool holders respectively. The Mori Seiki SH machining center was donated through MTTRF by Mori Seiki Japan. References [1] S.A. Tobias Machine tool vibrations Blackie London [] F. Koenigsberger J. Tlusty Machine Tool Structures vol. I: Stability Against Chatter Pergamon Press New York [] H.E. Merrit Theory of self-excited machine tool chatter Transactions of ASME 87 (1965) 7 5. [] J. Tlusty F. Ismail Basic non-linearity in machining chatter CIRP Annals (1981) 1 5. [5] S. Smith J. Tlusty Efficient simulation programs for chatter in milling CIRP Annals (1) (199) [6] D. Montgomery Y. Altintas Mechanism of cutting force and surface generation in dynamic milling ASME Journal of Engineering and Industry 11 () (1991) [7] I. Minis T. Yanushevsky R. Tembo R. Hocken Analysis of linear and nonlinear chatter in milling CIRP Annals 9 (199) [8] Y. Altintas E. Budak Analytical prediction of stability lobes in milling CIRP Annals (1) (1995) [9] T. Arvajeh F. Ismail Machining stability in high-speed drilling-part 1: modeling vibration stability in bending International Journal of Machine and Tools Manufacture 6 (1 1) (6) [1] T. Arvajeh F. Ismail Machining stability in high-speed drilling part : time domain simulation of a bending torsional model and experimental validations International Journal of Machine and Tools Manufacture 6 (1 1) (6) [11] J.C. Roukema Y. Altintas Kinematic model of dynamic drilling process IMECE-59 ASME International Mechanical Engineering Congress Anaheim CA. [1] J.C. Roukema Y. Altintas Time domain simulation of torsional axial vibrations in drilling International Journal of Machine and Tools Manufacture 6 (15) (6) [1] J.C. Roukema Y. Altintas Generalized Modeling of Drilling Vibrations Part I: Time domain model of drilling kinematics dynamics and hole formation International Journal of Machine and Tools Manufacture in press doi:1.116/j.ijmachtools [1] J.C. Roukema Mechanics and dynamics of drilling Ph.D. Thesis The University of British Columbia Vancouver Canada 6. [15] D.F. Galloway Some experiments on the influence of various factors on drill performance Transactions of ASME 79 (1957) [16] P.V. Bayly S.A. Metzler K.A. Young J.E. Halley Analysis and simulation of radial chatter in drilling and reaming DETC/VIB859 ASME Design Engineering Technical Conference 1999 Las Vegas Nevada. [17] P.V. Bayly S.A. Metzler A.J. Schaut S.G. Young Theory of torsional chatter in twist drills: model stability analysis and composition to test ASME Journal of Manufacture Science and Engineering 1 () (1) [18] G. Tlusty Manufacturing Processes and Equipment Prentice-Hall Englehood Cliffs NJ [19] P.V. Bayly J.E. Halley K.A. Young Tool oscillation and the formation of lobed holes in a quasi-static model of reaming DETC/ VIB861 ASME Design Engineering Technical Conference 1999 Las Vegas Nevada. [] P.V. Bayly K.A. Young J.E. Halley S.G. Calvert Analysis of tool oscillation and hole roundness error in a quasi-static model of reaming ASME Journal of Manufacture Science and Engineering 1 () (1) [1] P.V. Bayly M.T. Lamar S.G. Calvert Low-frequency regenerative vibration and the formation of lobed holes in drilling ASME Journal of Manufacture Science and Engineering 1 () () [] D.N. Dilley Accuracy vibration and stability in drilling and reaming D.Sc. Thesis Washington University St. Louis. [] S. Ema H. Fujii E. Marui Chatter vibration in drilling ASME Journal of Engineering and Industry 11 () (1988) 9 1. [] Altintas Y. Weck M. Chatter Stability in Metal Cutting and Grinding Annals of CIRP Key Note Paper of STC-M vol. 5/ pp