Chatter Stability of Metal Cutting and Grinding...sMs CsKs sj2,, GjH Φ =++→=ω Φω= ω+ ω (6) and cutting coefficient (K r) is assumed to be constant, the chatter stability of

Chatter Stability of Metal Cutting and Grinding

Y. Altintas1 (1), M. Weck2 (1) 1Manufacturing Automation Laboratory, Department of Mechanical Engineering

The University of British Columbia, Vancouver, Canada 2Laboratory of Machine Tools and Production Engineering, Aachen, Germany.

Abstract This paper reviews fundamental modeling of chatter vibrations in metal cutting and grinding processes. The avoidance of chatter vibrations in industry is also presented. The fundamentals of orthogonal chatter stability law and lobes are reviewed for single point machining operations where the process is one dimensional and time invariant. The application of orthogonal stability to turning and boring operations is presented while discussing the process nonlinearities that make the solution difficult in frequency domain. Modeling of drilling vibrations is discussed. The dynamic modeling and chatter stability of milling is presented. Various stability models are compared against experimentally validated time domain simulation model results. The dynamic time domain model of transverse and plunge grinding operations is presented with experimental results. Off-line and real-time chatter suppression techniques are summarized along with their practical applications and limitations in industry. The paper presents a series of research topics, which have yet to be studied for effective use of chatter prediction and suppression techniques in industry. Keywords: Cutting, grinding, chatter

1 INTRODUCTION Dynamics of metal cutting and grinding processes have been a focus area of manufacturing research since the establishment of CIRP. This article is dedicated to the memory of Professor J. Tlusty who contributed significantly to the understanding and engineering of dynamic cutting, stability and avoidance of chatter vibrations in machine tools. This article is compiled to review the current state of the knowledge in dynamic cutting, grinding and the existing research challenges in modelling and avoiding machine tool vibrations. Tlusty presented the overview of dynamic cutting in the last CIRP key note paper dealing with the topic in 1978 [102]. He focused extensively on the modelling and measurement of dynamic cutting coefficients, and their influence on the chatter stability in single point metal cutting processes. Peters et al. surveyed the grinding process models in 1984 [75], and Inasaki and Karpuschewski [47] recently compiled the state of the research on grinding chatter in a CIRP key note article. They focused on the modelling of grinding process dynamics, chatter stability and monitoring. Rivin presented a CIRP key note paper on tool-spindle interface dynamics [82]. He discussed the influence of structural interfaces on the dynamic stiffness of the machines in detail along with comparison of various design solutions. Weck reviewed the structural dynamics of recent Parallel Kinematic machine tool structures, and their comparison with Cartesian machine tools [119]. He summarized the assessment of the dynamic stiffness and performance of the PKM machine tools. Van Lutterwelt et al [108] and Byrne et al [18] reviewed the machining research studied over the years, and outlined most common metal cutting mechanics models used in predicting the cutting forces. While Finite Element based metal cutting simulation models are most common in analysing the plastic deformation trends at the cutting edge, orthogonal to oblique cutting transformation and mechanistic models are mainly used in predicting cutting forces exciting machine tool vibrations [2]. The CIRP key note articles listed above provide in detail and state of knowledge in cutting process mechanics [18], dynamic

cutting force coefficients [102], dynamics of grinding [47], machine tool design [119] and tool – spindle interfaces [82] which are most related to machine tool vibrations overviewed in this article. These authors provided a significant review of literature in the related areas, which will not be repeated here. However, there has been significant research progress in developing more advanced models in representing the dynamics of various cutting operations. With the advances in computer, sensor and high-speed machine tool technology, there have been new methods in predicting and avoiding chatter vibrations on the production floor. This article reviews the mathematical models of dynamic cutting and grinding, prediction of chatter stability for various operations, and off-line and on-line chatter avoidance techniques successfully used in the laboratories and industry. The article is organized as follows. The pioneering chatter stability theories of Tlusty [104] and Tobias [106] are presented in Section 2. Their theories provide fundamental understanding of dynamic cutting and chatter stability lobes. The application of orthogonal cutting chatter stability to single point machining operations, such as turning and boring is presented in Section 3. The current unsolved issues, such as dynamic cutting force coefficients and process damping, as well as the non-linearities in the stability models are discussed as research challenges. Section 4 reviews the dynamics and research challenges in predicting both forced and self-excited, chatter vibrations in drilling operations. The mathematical model of dynamic milling is presented in section 5. Past and recent stability theories in milling are reviewed, and the stability lobes predicted by various approaches are compared against exact, time domain, numerical solutions. Time domain modelling of dynamic grinding processes is presented in Section 6. Section 7 covers the review of on-line and off-line chatter vibration suppression techniques. A brief overview of challenging research tasks in dynamic cutting are listed in section 8, followed by a summary of the current state of knowledge in the field of dynamic cutting and grinding in section 9.

2 DYNAMICS OF ORTHOGONAL METAL CUTTING

AND CHATTER STABILITY LOBES The pioneering orthogonal chatter stability models were introduced by Tlusty and Polacek [104], and Tobias and Fiswick [106] almost at the same period but independent of each other. The basics of the one-dimensional chatter theory of Tlusty and Tobias are presented using an orthogonal, plunge turning operation shown in Figure 1a [2]. A disk held by a flexible shaft is connected to chuck and tail stock. The tool has a flat face, and the process produces tangential cutting forces (Ft) in the direction of cutting speed (V), and feed forces Fr in the radial, chip thickness direction that is perpendicular to the cut surface. If the system is rigid without experiencing any vibrations, the cutting forces are expressed as a linear function of width of cut ( a ) and the feed per revolution or static chip thickness (h0):

0 0,t t r rF K ah F K ah= = (1)

where Kt , Kr are the tangential and radial cutting coefficients, respectively [2]. The cutting coefficients Kt , Kr may be dependent on the tool geometry, chip thickness, cutting speed and lubrication environment depending on the work material and application. The turning system may experience transient vibrations at the first tool contact, and the tool leaves a wavy surface behind. If the vibrations do not diminish during the second revolution, the tool cuts dynamically changing chip thickness, which is mainly affected by the radial cutting forces. The general dynamic chip thickness formed due to regeneration of waviness on both sides of the chip surface is expressed as follows:

( ) ( ) ( )0h t h y t y t T⎡ ⎤= − µ − −⎣ ⎦ (2)

where T is the spindle rotation period, µ is the overlap factor, y(t) and y(t – T) are the present and past vibration amplitudes in the radial direction, respectively. The overlap factor is unity in the plunge cutting operation presented in this example ( 1µ = ), zero in threading ( 0µ = ) and can vary in cylindrial turning operations ( 0 1≤ µ ≤ ). By modeling the shaft -disk as a single degree of system with a lumped mass (m), stiffness (k) and damping (c) at the cutting point, the dynamic orthogonal cutting system becomes:

( ) ( ) 0

( )r r

r

My Cy Ky F K ah t

K a h y t y t T

+ + = =

⎡ ⎤= − − −⎣ ⎦

&& & (3)

The dynamic cutting equation (Eq.3) is a delayed differential equation which can be described by the block diagram given by Merrit [64] as shown in Figure 1. The transfer function of the closed loop system is derived as:

0

( ) 1( ) 1 (1 ) ( )sT

r

h sh s e K a s−

=+ − Φ

(4)

whose characteristic equation determines the chatter stability conditions,

1 (1 ) ( ) 0sTre K a s−+ − Φ = (5)

The structural dynamics of the set-up is expressed by its Frequency Response function (FRF) as:

( )( ) ( ) ( )

2 , ,s Ms Cs Ks s j

G jH

Φ = + + → = ω

Φ ω = ω + ω (6)

and cutting coefficient (Kr) is assumed to be constant, the chatter stability of the system is only dependent on the depth of cut (a) and spindle period (T). The machining system will be stable, critically stable or unstable depending on the roots of the characteristic equation (Eq.5). The system is critically stable when the real part of the root is zero. In other words, the vibrations neither grow exponentially nor diminish, but are sustained with equal amplitude (y(t) ≡ y (t –T)) at a chatter vibration frequency ( cω ). Both Tlusty [104] and Tobias [106] independently formulated the following absolute chatter stability law which has been widely used since 1950s:

lim1

2 ( )ra

K G−

=ω

(7)

The stability equation leads to positive real depth of cut only when the real part )(ωG of the transfer function between the tool and workpiece is negative. Eq. (7) gives only absolute depth of cut when the minimum value of )(ωG is considered as shown in Figure 2. However, the characteristic equation of dynamic cutting (Eq.5) has an infinite number of solutions for each possible spindle period (T ). Tobias utilized this property [106], and introduced the speed dependent chatter stability lobes as:

Figure 1: Dynamics of orthogonal cutting system [64].

1 ( )tan , 3 2( )

2 602 c

HG

kT nf NT

− ω ⎫ψ = ε = π + ψ ⎪ω ⎪

⎬π + ε ⎪= → =⎪π ⎭

(8)

where , , ck fε are the phase shift between the inner and outer waves, number of vibration waves or lobes generated in one spindle period (T ), and chatter vibration frequency ( cf ) in [Hz], respectively. The chatter stability lobes are constructed by scanning the possible chatter frequencies from the transfer function where the real part is negative, e.g. ω <( ( ) 0)G , see Eq. (7). When the depth of cut and spindle speed are selected under the stability lobe, the process would be stable leading to smooth surface finish and less dynamic loads on the machine tool. If the cutting conditions are above the lobe, the process becomes unstable with growing vibrations, high dynamic forces and rough surface finish. The chatter frequency can be detected using the Fourier Spectrum of vibration signals or from the wave length of the marks left on the surface [46].

Tobias pointed out that the cutting coefficient Kr could be more accurately modeled as a complex number which includes the changes in the effective rake angle of the tool, the slopes of the waves left on the inner and outer surfaces of the chip, and process damping created by the friction between the clearance face of the tool and wavy surface generated on the work surface [43][102], see Figure 3. The stability of metal cutting operations at low spindle speeds which are an order of magnitude lower than the chatter vibration frequencies are still difficult to model and predict mainly due to difficulty in modelling the process damping. When there are a significant number of vibration waves left on the surface during one revolution of spindle in turning or in one tooth period of milling, the effective clearance angle of the tool becomes zero or negative as seen in Figure 3. The clearance face of the tool rubs against the waves causing friction forces against the direction of motion, hence damping out the chatter vibrations. Kegg showed that the process damping or friction between the wavy surface of the workpiece and flank is most crucial in determining the chatter stability at low speed cutting conditions [53][89] where the ratio of spindle period to chatter vibration period is high, i.e. at the lobes which are higher than five ( 5>k ). The process

damping is most effective when the spindle speed is low relative to the chatter vibration frequency. However, there is no mathematical model which considers the interaction between the tool’s clearance face and the wavy surface. In order to consider the process damping in chatter stability, CIRP formed a task force identifying the dynamic cutting coefficients using a common test rig proposed by Peters, Vanherck and van Brussel [72]. Tlusty from VUOSO-Prag and McMaster University, Vanherk, Peters and van Brussel from Katholieke University of Leuven, Weck from WZL TH Aachen and others participated in the study. Tlusty summarized the research findings in great detail in the 1978 CIRP key note article [102]. The dynamic cutting force coefficients were calibrated experimentally by correlating the tool geometry, chatter vibration frequency and cutting speed. However, the experimental rigs were limited to low vibration frequencies and cutting speeds, and the experimental procedure was too cumbersome to be used in practical production applications. Montgomery and Altintas attempted to model the penetration of a hard tool into the softer work material with wavy surface finish using laws of contact mechanics [67]. However, the contact model produced high frequency bouncing of the tool due to a poor prediction model of the complex contact mechanism in the proposed time domain simulation system. Modelling of process damping as a function of tool geometry, tool wear, cutting speed, work material and vibration frequency is yet to be developed for effective prediction of chatter vibrations and stability in turning, boring, and low speed milling and drilling operations [33], [101]. The basic chatter theory presented by Tlusty and lobes introduced by Tobias had fundamental impact in designing machine tools and the selection of productive cutting conditions, and led to expanded stability formulations and interpretations [70], [78], [117]. Although the theory was based on orthogonal cutting conditions, where the machining system is continuous without time varying dynamics and excitation, researchers and engineers used it with great success by simple adjustments based on intuitions and experience. 3 DYNAMICS OF TURNING AND BORING

OPERATIONS While the boring bar is usually the most flexible part in hole enlargement operations, the shaft, chuck, tailstock and tool holder may contribute to the flexibility which leads to chatter in turning operations. However, both operations have similar mechanics and dynamics due to having geometrically defined cutting tool edges. A general diagram of a single point cutting operation is shown in Figure 4 It is customary to model the cutting

Figure 2: Stability lobes based on orthogonal chatter theory.

Figure 3: Process damping mechanism in dynamic cutting, after Tlusty [103].

forces in oblique cutting coordinates [2], i.e. tangential or cutting speed direction, (Ft), chip thickness direction or perpendicular the cutting edge (Ff), and along the cutting edge (Fr) as shown in Figure 4 The cutting forces in oblique coordinates are expressed as:

, ,t t f f r rF K bh F K bh F K bh= = = (9)

where the cutting force coefficients (Kt, Kf, Kr) can be evaluated either from shear stress, shear angle, friction coefficient and tool geometry, or by mechanistic curve fitting to experimental force data [2]. Alternatively, some researchers use the resultant cutting force [117] which can be constructed from the oblique cutting forces from Figure 4 as:

( ) ( )

( ) ( )( )( )

2 2

2 2

1

1

1 / /

cos , cos 1/ 1 / /

cos tan , tan /

cos tan , tan /

c t f t r t c

t c f t r t

f c a a f t

r t r r r t

F bhK K K K K K bh

F F K K K K

F F K K

F F K K

−

−

= + + =

= γ → γ = + +

= γ β → β =

= γ β → β =

(10)

The cutting forces are projected to the machine tool coordinate system as:

0 sin cos cos1 0 0 cos cos

0 cos sin cos cos

x r r

y a c

r r rz

FF F

F

⎧ ⎫ − ψ − ψ γ⎡ ⎤ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎢ ⎥= − γ β⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥− ψ ψ γ β⎣ ⎦ ⎩ ⎭⎩ ⎭

(11)

It is assumed that the system is flexible in all three machine coordinates, and the vibrations ((x, y, z) can be estimated from the cutting forces (Fx, Fy, Fz) and the measured or predicted Frequency Response Function Matrix Φ⎡ ⎤⎣ ⎦ as:

[ ] , xx xy xz x

yx yy yz y

zx zy zz z

x Fy F X Fz F

⎡ ⎤Φ Φ Φ⎧ ⎫ ⎧ ⎫⎢ ⎥⎪ ⎪ ⎪ ⎪= Φ Φ Φ → = Φ⎨ ⎬ ⎨ ⎬⎢ ⎥

⎪ ⎪ ⎪ ⎪⎢ ⎥Φ Φ Φ⎩ ⎭ ⎩ ⎭⎣ ⎦

(12)

The resulting vibrations must be projected in the direction of chip thickness which is perpendicular to the cutting edge, so that the deformed chip thickness can be evaluated:

sin 0 cos Td r rh x y z= ψ ψ (13)

By combining Equations (10)

, ( , , , )d o c o pq pqh F d p q x y z= Φ Φ = Φ → =∑ 14)

where Φo is called Oriented Transfer Function, and the

pqd parameters are called directional factors which are given below for a stationary cutting edge with an approach angle of rψ :

( )

( )

( )

( )

2

2

2

2

cos 0.5 tan sin2 tan sin

0

cos 0.5 tan sin2 tan cos ;

cos sin , 0, cos cos ;

cos 0.5 tan sin2 tan sin

0

cos 0.5 tan sin2 tan cos

xx r r a r

yx

zx a r r r

xy r yy zy r

xz a r r r

yz

zz r r a r

d

d

d

d d d

d

d

d

= − γ β ψ + β ψ

=

= − γ β ψ + β ψ

= − γ ψ = = − γ ψ

= γ − β ψ + β ψ

=

= γ + β ψ − β ψ

(15)

The directional factors are used to account for the variations in the chip thickness due to vibrations in machine coordinates that are excited by the cutting forces. Tlusty [56], Peters [73], Weck [117] and others applied the real part of the oriented transfer function ( 0Φ ) to the orthogonal chatter stability given in Eq. (7) and (Eq.8), and obtained chatter stability lobes for single point machining operations [50],[126] as :

( )1

lim 2b

K Rec o= −

Φ (16)

However, the accuracy of the chatter prediction was not always satisfactory due to several factors [62]. The chatter vibration frequency is typically above fc is greater than 200[Hz] in single point machining operations depending on the bar length in boring or the dimensions of the shaft to be turned, while the spindle speed is under 1500 rev/min or n is less than 25[Hz]. The integer ratio of chatter frequency over the spindle frequency gives the location of the lobe ( k ) where the cutting takes place,

int( ), 2 ( )c cf fk fracn n

= ε = π •L (17)

and ε is the phase shift between the waves left on the surface during two subsequent revolutions. Since the cutting takes place at higher lobes where the spindle speed is low in single point cutting operations, the operation is always in the process damping region where

Figure 4 : Directional factors for a single point tool.

Figure 5. The cutting force directions and thus the cutting coefficients change as a function of feed rate, nose radius and depth of cut in boring and

turning [11].

the flank interferes with the wavy surface leading to friction as shown in Figure 3. In addition, the flank friction changes at every point on the wave, causing the damping to be harmonically varying as a function of tool angles, vibration frequency and cutting speed. The satisfactory prediction of chatter stability of boring and turning in frequency domain needs modeling of process damping in terms of dynamic or complex cutting coefficients as discussed in the previous section [102].

In addition to process damping, the boring and turning processes have nonlinear dynamics [27]. The chip thickness distribution along the cutting edge is dependent on the radial depth of cut, feed rate in the axial direction, nose radius and approach angle [10]. When the tool has a nose radius, there isn’t any constant approach angle, and therefore the average direction of the cutting force depends on the feed rate, nose radius, approach angle and radial depth of cut [11][12], hence Eq. (7) cannot be used directly, see Figure 5. The frequency domain solution of such a non-linear system can be solved only by linearizing the system around a narrow band of depth of cut and feed, which is not an ideal solution. The researchers developed time domain solutions in predicting the chatter stability. Opitz [71] used analog computers, and Tlusty [100] simulated the dynamic turning processes by converting the (Eq.3) into difference equations using a sampling period 8-10 times higher than the highest natural frequency participating in chatter vibrations. Sato [57],[85] pointed out the importance of considering multiple regenerations in predicting the stability of turning, since the separation of tool and workpiece due to excessive vibrations bring saturation and nonlinearity to the process. Kuster [60], Lazoglu et al. [61] proposed time domain simulation algorithms in predicting dynamic cutting forces, regeneration of waviness, vibration frequency and amplitude in boring operations. Lazoglu used true kinematics of both rigid body and structural dynamic motions of boring, as well as the loss of tool contact and nonlinearity in cutting force coefficients. However, the time domain simulation of boring and turning is extremely time consuming due to the necessity of digitizing the high number of waves left on the turned/bored surface which is kept in the memory for accurate evaluation of dynamic chip thickness and multiple regeneration. Even the frequency domain solution needs to consider multiple regenerative feedback [62][57] where each wave may have a different overlap and cutting force coefficient due to non-linear dependency of the cutting forces on depth of cut and chip load, see Figure 6. In short, there is still significant research to be done in predicting the chatter stability in

single point machining operations where the spindle to chatter frequency ratios are very low. 4 DYNAMICS OF DRILLING

A drill is a rotating, flexible, pre-twisted beam with cutting lips at its free end, where material is sheared away to generate a hole, see Figure 7. The center of the drill, the chisel edge, extrudes the material by indentation. A pilot hole can be used to reduce the influence of the chisel edge thus the thrust force at the tip, which may reduce the wandering of the drill [126]. The equations of motion for the dynamic drilling system can be formulated as follows:

&& &

&& &

&& &

&& &

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

x

y

z

c

Fx t x t x tFy t y t y t

M C KFz t z t z tTt t t

⎧ ⎫⎡ ⎤ ⎧ ⎫ ⎡ ⎤ ⎧ ⎫ ⎡ ⎤ ⎧ ⎫⎪ ⎪⎢ ⎥ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ + =⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪θ θ θ⎣ ⎦ ⎩ ⎭ ⎣ ⎦ ⎩ ⎭ ⎣ ⎦ ⎩ ⎭ ⎩ ⎭

(18)

where yx, denote the lateral deflections, z the axial and θ the torsional deflection of the drill, as illustrated in Figure 7. The matrices M, C, K reflect the mass, damping and stiffness characteristics at the drill tip, respectively. The external load acting on the drill include two lateral (Fx, Fy) and thrust (Fz) forces, and a torque (Tc). Galloway [38] presented the mechanics of drilling which leads to the prediction of cutting forces and torque. However, in dynamic drilling, the forces and torque are a function of the chip thickness, width of cut, material properties, drill geometry, drill tip vibrations at the current time (t) and

Figure 7: Deformed drill showing lateral deflections x,y; axial deflection z and torsional

deflection θ (Source : MAL UBC).

Figure 6: Block diagram of boring with multiple regenerative feedback [61].

one tooth period earlier )( τ−t . When the drill vibrates, one has to identify the chip thickness distribution along the cutting edge by considering vibrations in all directions in order to predict the cutting load. xF is given as an example here:

( ) , , ( , ),, ( ), ( )x xF F x y z t t t t= − τ θ θ − τ (19)

Since the cutting forces are dependent on vibrations in all directions, it is challenging to solve the equation of motion (Eq.19) for the dynamic drilling. Furthermore, the process mechanics are complicated because of the chisel’s ploughing action, and the cutting lips have varying rake angle and chip thickness along its edge.

Although the complete dynamics of drilling have not been fully developed, some noted research has been conducted by considering individual vibration mechanisms. Galloway [38] experimentally studied bending vibrations that cause out of round holes, and associated the form errors to the drill asymmetry. Gupta et al. [41][42] used their model to predict how the hole profile changes along the axial depth of cut. Reinhall and Storti used a simple analytical model that predicts the polygonal hole profiles formed in drilling through thin plates [79]. Tekinalp, Rincon and Ulsoy studied the bending and torsional vibrations of drills via finite element analysis [80][98][99]. Although the gyroscopic effects were included at high spindle speeds, they did not model the interaction between the torsional and bending vibrations of the drill with the cutting forces. One specific source of chatter in drilling is torsional-axial coupling of drill vibrations. As the drill vibrates in torsion, it lengthens and shortens, leaving a wavy surface on the bottom of the hole. The resulting wavy surface causes time-delayed, regenerative torque and thrust force on the tool. Depending on the phase difference between the current tool motion and the prior surface, and the radial depth of cut, the oscillatory force grows and the vibrations become unstable. If the vibration amplitudes are large, then part of the tool may lose contact with the material being cut, which is a nonlinearity in the process. Sample hole surfaces generated by torsional-axial chatter vibrations are shown in Figure 8.

4.1 Torsional-axial chatter stability

Bayly et al. [15] studied the stability of the drilling process by ignoring lateral vibrations, but considering only torsional and axial displacements. They neglected the chip thickness variation due to torsional vibrations. The dynamic drilling process is thus reduced to the following

one dimensional equation of motion in the axial direction:

( ) ( ) ( )f t cmz cz kz K K Ra bN z t z t+ + = − + − − τ&& & (20)

where b is the radial width of cut, N is the number of flutes, and

fK and tK are the thrust force and torque

coefficients, respectively. The system is thus still excited by torque and thrust, which both depend on the axial deflection at the current time )(tz , and one tooth period earlier ( )z t − τ . The torsional-axial coupling parameter ac relates the applied torque to excitation in the torsional-axial mode, and is determined from modal analysis of the drill. The radial width of cut is the difference between the drill radius and the pilot hole radius. Bayly et al. [15] applied Tlusty’s one- dimensional stability law and found the critical width of cut as:

12 ( )Re ( )f t c c

bN K K Ra

−=

+ Φ ω (21)

4.2 Lateral chatter stability

Bayly et al. [16] also studied the case where torsional-axial vibrations are ignored, and chip thickness regeneration due to lateral vibrations is considered. In order to simplify the expression of the cutting forces, the equations of motion for a two-fluted drill are formulated in rotating coordinates u, v. (Figure 9) illustrates the following coordinate transformation used:

cosθ sinθsinθ cosθ

xuu yv⎡ ⎤ ⎧ ⎫⎧ ⎫⎪ ⎪ ⎪ ⎪⎢ ⎥⎨ ⎬ ⎨ ⎬

⎪ ⎪ ⎪ ⎪⎢ ⎥⎩ ⎭ ⎩ ⎭⎣ ⎦= =

− (22)

The equations of motion in the rotating coordinate system are then as follows:

2

2

22

x x x

y y y

ux x x

vy y y

m c mu um m cv v

Fk m c uFc k m v

&& &

&& &

− Ω+ +

Ω

− Ω − Ω=

Ω − Ω

⎡ ⎤ ⎡ ⎤⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥⎩ ⎭ ⎩ ⎭⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎧ ⎫⎧ ⎫⎨ ⎬ ⎨ ⎬⎢ ⎥⎩ ⎭ ⎩ ⎭⎣ ⎦

(23)

0 ( ) ( ) ( ) ( )0 ( ) ( ) ( ) ( )

u tc

v r

F K u t u t u t u tK

F K v t v t v t v t+ − τ + − τ

= =+ − τ + − τ

⎧ ⎫ ⎡ ⎤ ⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎢ ⎥

⎩ ⎭ ⎩ ⎭⎩ ⎭ ⎣ ⎦

Where Kt and Kr are the tangential and radial cutting force coefficients, respectively. The dynamic properties of the drilling system are identified in the fixed coordinate

Figure 8: (a) chatter surface on the bottom surface of a hole for a twist drill (with pilot hole); (b) for an indexable

drill (Source: UBC – M.A.L.).

Figure 9: Definition of rotating coordinate system in drilling.

system, in two orthogonal directions x, y. The dynamic properties in x direction are for example described by only one mode, characterized by , ,x x xm c k . The regenerative cutting force depends on the current deflection ( )u t and one tooth period ( τ ) earlier, ( )u t − τ . To analyze the chatter behavior, a harmonic solution 1 2

T i tu v U U e ω= is assumed. Substituting this

into the equations of motion (Eq.(23)), the condition for the existence of such a solution becomes:

[ ( , ) (1 )] 0icI b K e U− ωτ+ Φ ω Ω + = (24)

where 2 1( , ) [ ( ) ( )]M i C K −Φ ω Ω = −ω + ω Ω + Ω is the frequency response matrix of the system in the rotating frame. Due to the presence of a zero column in the cutting stiffness matrix

cK , the following scalar characteristic equation of dynamic drilling is obtained:

− ωτ+ Φ ω Ω +Φ ω Ω + =1 ( ; ) ( ; ) (1 ) 0ixx x xy yb k k e (25)

where xxΦ and xyΦ are components of the frequency

response matrix of the system in the rotating frame. Note that a “+” sign appears in the term (1 )ie ωτ+ because in a rotating frame, the displacements of opposing teeth have opposite signs. For a given speed Ω and corresponding time delay /τ = π Ω , the chatter frequency cω and limiting radial depth of cut b are sought. The critical radial depth of cut is then found from Tlusty’s chatter stability theory as:

12Re( )xx t xy r

bK K−

=Φ + Φ

(26)

4.3 Lateral whirling vibrations

Bayly, Lamar and Calvert [14] studied lateral whirling vibrations by assuming that the mass and damping terms can be neglected at low spindle speeds. Only the drill flexibility is considered which makes the process a quasi-static system. The equations of force equilibrium for a two-fluted drill in the rotating frame are expressed as a

discrete-time matrix equation describing successive states of the tool. Cutting and rubbing forces acting on the tool are regenerative in nature. Guessed process damping forces are added for the chisel edge region. The eigenvalues and eigenvectors determined from the state transition matrix yield an exponential solution of the tool motion. When drilling a full hole, a vibration of slightly less than 3 cycles per revolution is obtained, so this is a backward whirling motion. When cutting, rubbing and process damping forces at the chisel edge are removed from the model, thus simulating the drilling of a piloted hole, the least stable mode is found to be a little less than 7 cycles per revolution, which is also a backward whirl. Bayly’s results are illustrated in Figure 10, along with experimental measurement of one revolution for each case. 4.4 Dynamics of deep hole drilling Deep hole drilling methods are used when machining bore holes with a high length to diameter ratio. In contrast to conventional twist drills, deep hole drilling tools have an asymmetric cutting edge arrangement resulting in a non zero radial cutting force component. In combination with guide pads, this leads to reduced run-out error and better surface finish. According to the type of cooling lubricant supply [69], deep hole drilling methods can be classified into gun-drilling, ejector- and BTA (Boring and Trepanning Association) -drilling. The working principle of BTA drilling, incorporating external cooling lubricant supply and internal chip removal through the boring bar, is illustrated in Figure 11. This method is usually employed for machining bore holes with 20 mm diameter and above. Due to the slender tool-boring-bar assembly, where the length to diameter ratios of up to 100 are common for deep bores, the gun drilling process is highly susceptible to vibrations. Chatter vibration in deep hole drilling occurs mainly due to torsional vibration of the tool-boring bar assembly. Analyses and time invariant models for gun drilling and BTA-drilling were presented by Streicher [94], Thai [96], Chin [24] and Weber [120]. A detailed investigation of the dynamic behavior of the bar in BTA-drilling was carried out by Chin et al. [24]. Weinert et al. [121][122] have studied the development of the process dynamics by analyzing the drilling torque signals recorded during the process. They observed that chatter and non-chatter phases may alternate, and the modes involved in chatter vibration may change during the process. Based on the analysis of the temporal evolution of the spectral power of the torque signals at the relevant frequencies, it was shown that a long term prediction of the phase changes is possible. A non-linear oscillator based on the van der Pol equation [107] was suggested as a starting point for modeling the dynamics. For chatter suppression in gun-drilling, a damping system at the interface between the tool shaft and the spindle was proposed by Streicher [94] and different design variations were later

Figure 10: Simulated and experimental hole profile; (a),(b): full hole (3 sided); (c),(d): piloted hole (7 sided),

after Bayly et al. [14].

Figure 11: Principle of BTA deep hole drilling [69].

implemented and evaluated by Hauger [44]. Thai [96] suggested the application of a Lanchester damper coupled with the boring bar. Based on this idea, an automatically controlled damping system was later developed and implemented by Bolle [17] for BTA drilling operations. As one of the earliest discussions, Kronenberg [58] argued that the whirling vibrations are either caused by the bending modes of the boring bar or the static unbalance of the forces acting on the tool head. According to Pfleghar [76], the latter is the most significant source of whirling vibrations in gun-drilling. He defined a stability measure for evaluating the force system and used it to optimize the positioning of the guide pads. Stockert [95] evaluated the effect of the tool design parameters on this disturbance for BTA drilling. He furthermore differentiated three different types of whirling vibration according to their point of occurrence: at the start of the process, reproducibly at the same drilling depth and seemingly at random. For the second type of occurrence, Gessesse et al. [39] have shown a relationship between the bending modes of the boring bar and whirling vibration. The effect is most likely to occur when an odd-numbered multiple of the tools rotational frequency coincides with an eigen-frequency associated with the bending mode of a boring bar. The bending modes have to be considered as being dependent on the current drilling depth due to continuously changing support conditions during the process, which leads to significant frequency shifts [40]. 4.5 Research challenges Significant amount of research effort has been spent on modeling the mechanics of drilling to predict the cutting forces accurately. Despite such efforts, the chisel edge region and the various modifications to it that are used in practice complicate the process models. The significant changes in cutting speed, rake and oblique angles along the cutting edge make accurate prediction of cutting forces in the chisel region difficult. Several aspects of drilling vibrations have been studied, focusing on single modes of vibrations and explaining each phenomenon with a different model. In reality lateral, torsional and axial vibrations occur simultaneously when drilling a hole. The entry and exit transient parts of the whole operation can only be modeled using time domain simulation techniques which consider the coupled dynamics of drilling by including torsional, axial and lateral vibrations simultaneously. The mechanics of chatter vibration in deep hole drilling have so far been restricted to time invariant models. Time series analyses and modeling of torque data from BTA drilling test show that the process is strongly time- or drilling depth dependent, which must be considered in more refined mathematical models of the process. Whirling vibration in deep hole drilling is still difficult to predict and its occurrence usually leads to a significantly damaged work piece. The development of real-time avoidance techniques for whirling vibration in deep hole drilling is also still an area of interest for industry. 5 DYNAMICS OF MILLING Milling belongs to the class of multi-point machining operations where the process is intermittent and periodic at tooth passing intervals. There has been significant research advances in milling during recent years, especially due to its importance in high speed machining of dies, molds and monolithic parts from solid blocks. Tlusty and his school of co-workers have contributed to an in-depth understanding of milling mechanics, chatter and its prevention. Additional researchers provided

complimentary methods and insight to the dynamics of the milling process, which all together led to successful utilization of knowledge in industry with significant productivity gains. The fundamental theories of chatter in milling are summarized here.

5.1 Mathematical modeling of dynamic milling process-one dimensional model

A simple model of a milling process is considered here as shown in Figure 12. Assume that there is only one tooth which is at the radial immersion position φ measured clockwise from the y axis. The feed direction is aligned with the x axis of the machining system. Neglecting the axial force for simplicity, there are two rotating force vectors acting on the tooth, tangential force (Ft) and radial force (Fr), which are expressed as follows [3]:

1 1

( ),( ),

tan tan

t t

r r t r t

rr

t

F K ahF K F K K ah

F KF

− −

= φ

= = φ

θ = =

(27)

where Kr is the ratio of radial to tangential cutting forces. The resultant cutting force on the tooth becomes:

2( ) 1 ( ) ( )t r sF K K ah K ahφ = + φ = φ (28)

where the resultant cutting force coefficient is 21s t rK K K= + . When the tool has an approach angle, a

three dimensional model of the milling force must be considered similar to the turning system presented in Section 3, or with a more detailed milling model as presented by Altintas in [4]. The cutting forces in the feed and normal directions can be resolved as follows:

( ) cos( ), ( ) sin( )x yF F F Fφ = − φ − θ φ = φ − θ (29)

The dynamic chip load created by the tooth and vibrations is [3]:

[ ( )] sin ( ) ( )sin ( ) ( )cos ( )h t c t x t t y t tφ = φ + ∆ φ + ∆ φ (30)

where ( )t tφ = Ω is the angular position of the tooth for a spindle speed of Ω (rad/s). Note that the static chip load

sin ( )c tφ is an input to the closed loop dynamics of the chatter, and does not affect the critical stability of the

Figure 12: Dynamics of milling.

linear, dynamic machining system. Both vibration components ( , )x y∆ ∆ are dominated by the chatter vibration frequency )( cω ; so as the resultant cutting

force, e.g. ( ) ci tF t Fe ω= . The vibrations at present (t) and previous tooth (t-T) periods can be expressed by [2],

-( ) ( ), ( ) ( ) ( )0

-( ) ( ), ( ) ( ) ( )0

( ) - , ( ) -0 0

i Tcx i F x e i Fxx c x c c xx c x ci Tcy i F y e i Fyy c y c c yy c y c

x x x y y yc c

ω= Φ ω ω ω = Φ ω ω

ω=Φ ω ω ω = Φ ω ω

∆ ω = ∆ ω =

(31)

where xxΦ and yyΦ are direct frequency response functions (FRF) of the structure in x and y directions, respectively. Since the cutting forces Fx and Fy are both dependent on the vibrations in the directions (x, y), the system has coupled dynamics. However, Tlusty simplified the process by orienting the cutting forces from the directions of orthogonal springs to the direction of resultant cutting force as follows [101][105]:

( 1) cos( )

( 1) sin( )

i T i tc cx e Fexxi T i tc cy e Feyy

− ω ω∆ = + − Φ φ− θ

− ω ω∆ = − − Φ φ−θ

(32)

Substituting (Eq.32) in the dynamic chip thickness (Eq.30):

⎧ ⎫⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

Φω ω= φ φ θ φ φ θΦ

-( ) ( -1) sin cos( - ) -cos sin( - ) xxi T i tc ch t e Feyy

⎧ ⎫⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

Φω ω= φΦ

-( -1) ( ( ))xxi T i tc ce A t Fei yy

(33)

The formulation given here orients vibrations and cutting forces from x, y spring directions to the direction of chip load )(φ . ( ( ))iA tφ is a periodic function and valid only

between the start ( stφ ) and exit angles ( exφ ) of cut. Tlusty used geometric mean of the immersion angle [101], rather than taking an average value of ( ( ))A tφ as used by Opitz [71] and Weck [117] as :

0 2ex st

stφ − φ

φ = φ + (34)

The direction factors then become constant as:

0 0 0 0sin cos( ), cos sin( )x yu u= φ φ − θ = − φ φ − θ (35)

which leads to time invariant and constant oriented transfer function:

0,1 x xx y yyu uΦ = Φ + Φ (36)

However, unlike turning, the directional factors change as a function of spindle rotation, and they are periodic at

cutter pitch angle2pN

φ =π

. Opitz [71] used the average

of the periodic directional function of the resultant force as opposed to geometric mean adopted by Tlusty.

( ) ( )

1 exA sin cos cos sin d0 stp

exN 1 1 (sin ) cos(2 )2 2 4

stex1 1(sin ) cos(2 ) v vx y2 4st

φ ⎡ ⎤= φ φ−θ − φ φ−θ φ∫ ⎣ ⎦φφ

φ= θ φ − φ−θ

π φ

φθ φ+ φ−θ =

φ

(37)

The oriented transfer function of Opitz is also time invariant and constant, but different than Tlusty’s approach.

0,2 x xx y yyv vΦ = Φ + Φ (38)

Weck [117] further considered the influence of direct and cross transfer functions of the machine tool compliance similar to the method presented for turning in Section 3. However, while noting the time variation of the directional factors, he also averaged them and oriented all the vibrations at the cutting edge location. Hence, the time dependency from the chip thickness is still removed, and the chatter stability problem becomes a one-dimensional scalar problem since it is oriented in a fixed single direction like in turning. It can be solved by classical chatter theory presented earlier in 1950s by Tlusty [104] or Tobias [106]. The mean dynamic chip thickness becomes:

0( 1)c ci T i tmh e F− ω ω= − Φ (39)

The average of the periodic function corresponds to the average of the dynamic chip thickness, which leads to mean dynamic resultant cutting force,

0( 1)c c ci t i T i ts m sFe K ah K a e Feω − ω ω= = − Φ (40)

For critical borderline stability analysis, the characteristic equation of the dynamic milling becomes,

lim 01 (1 ) ( ) 0ci Ts ce K a i− ω+ − Φ ω = (41)

where lima is the maximum axial depth of cut for chatter vibration free machining. The stability lobes are then solved using the same formulation given for one dimensional theory [104][106]:

lim1

2 Re ( )2 60

2

s c e

c

aK m

kT nf NT

⎫⎪⎪⎬⎪⎪⎭

−=Φ ω

π+ ε= → =π

(42)

where cf [Hz] is the chatter frequency, T [s] is the tooth passing period, N is the number of teeth on the cutter, and n[rev/min] is the spindle speed. Re Φ is the real part of the oriented transfer function that can be evaluated by either approximations given in (Eq.36) by Faassen (36) or in (Eq.38) by Opitz. Tlusty [101] adjusted the stability limit by scaling the system by an average

number of teeth in cut, ( ) /(2 )e ex stm N= φ − φ π . Since Opitz and Weck used the average resultant force direction by considering the pitch angle, 1em = must be used in their models. The phase shift of the chatter waves can be found from:

13 2 . tan oo

HG

−ε = π+ ψ →ψ= (43)

The expression given in (Eq.42) has been widely used as an extension of orthogonal chatter theory applied to milling.

5.2 Mathematical modeling of two dimensional dynamic milling model

Sridhar et al. [93] argued that the milling has coupled dynamics, and must be modeled as at least two dimensional problem. Instead of orienting the dynamic resultant cutting forces and the vibrations in one average direction, they are resolved in feed (x) and normal (y) directions:

cos sinxj tj j rj jF F F= − φ − φ , sin cosyj tj j rj jF F F= φ − φ (44)

where j=0,1,..,N-1 is the tooth index number. Summing the cutting forces contributed by all (N) teeth, the total dynamic milling forces acting on the cutter are found:

1

0

N

x xjj

F F−

=

= ∑ , 1

0

N

y yjj

F F−

=

= ∑ (45)

where j pjφ = φ + φ , and cutter pitch angle

is 2 /p Nφ = π . Substituting the chip thickness (Eq.30) and tooth forces (Eq.27) into (Eq.44), and rearranging the resulting expressions in matrix form yields [3]:

12

xx xyxtc

y yx yy

a aF xaK

F a a y

⎡ ⎤⎧ ⎫ ∆⎧ ⎫⎪ ⎪ = ⎢ ⎥⎨ ⎬ ⎨ ⎬∆⎢ ⎥⎪ ⎪ ⎩ ⎭⎩ ⎭ ⎣ ⎦ (46)

where time varying directional dynamic milling coefficients are given as,

1

01

01

01

0

[sin2 (1 cos2 )]

[(1 cos2 ) sin2 ]

[(1 cos2 ) sin2 ]

[sin2 (1 cos2 )]

N

xx j j r jjN

xy j j r jj

N

yx j j r jjN

yy j j r jj

a g K

a g K

a g K

a g K

−

=

−

=

−

=

−

=

= − φ + − φ

= − + φ + φ

= − φ − φ

= φ − + φ

∑

∑

∑

∑

(47)

Considering that the angular position of the parameters change with the time and angular velocity, i.e.

( )j pt j tφ = φ + Ω , Eq.(46) can be expressed in time domain in a matrix form as:

1 ( ) [ ( )] ( )2 tF t aK A t t= ∆ (48)

The directional matrix [A(t)] is time varying as the cutter rotates, and also couples the vibrations in each direction with the machining process. [A(t)] is a periodic function at tooth passing frequency (ωt = n/(60N) ) and its harmonics may be increasingly stronger as the radial depth of cut becomes smaller, i.e. the degree of intermittency becomes stronger [19]. Sridhar formulated the dynamic milling shown in (Eq.48), but solved it using numerical techniques. Opitz [71] used an analog computer to solve it. Weck averaged each directional factor over a pitch angle and oriented the vibrations in fixed directions of the machine in order to apply Tlusty’s one dimensional chatter stability theory [114]. Weck argued that such an approximation is valid only when the fluctuating component of the directional factors is very low which occurs when the number of teeth is very high and radial depth of cut is large. Minis and Yanushevsky solved the problem analytically in frequency domain by applying Floquet theory for delayed differential equations [66]. They checked the stability of dynamic milling by scanning speeds and depth of cuts like in chatter experiments, and they did not consider harmonics of periodic directional matrix [A(t)]. Budak and Altintas [19][20] considered the harmonics by proposing a multi-frequency solution. Each harmonic (r) of the periodic directional matrix [A(t)] becomes:

( )2

ex

st

irNr

NA A e dφ

− φ

φ

= φ φ⎡ ⎤⎣ ⎦ π ∫ (49)

Davies et al. considered a special case where the radial depth of cut is very small [29]. They considered the time in cut and out of cut ratio, dynamics in one direction, and predicted small pockets of stability at high-speed range. Insberger and Stepan [48] approximated the regenerative time delay, and solved the stability of unidirectional milling. Their approximation was sufficient for high speed region, but inaccurate at low speeds due to approximated time delay. Bayly et al. solved the Eq. (48), using time finite elements [13]. Their method is somewhat similar to time domain solution of chatter but with improved computational efficiency. Multi-frequency solution of Budak and Altintas [19] and Merdol and Altintas [63], Davies’s solution for a system which has flexibility in one direction [29], Corpus and Endres’s high order solution for interrupted cutting [28] and Bayly’s time finite element solution [49] led to the prediction of added lobes which can not be obtained when the harmonics of the tooth passing frequency spread from the chatter frequency is neglected [3]. However, the objective of stability analysis is to predict the speeds and depth of cuts accurately but in a computationally efficient and practical way. Altintas et al. [8] argued that the harmonics are low pass filtered by the dynamics of the system if the radial immersion is not too small. The time variation of the directional matrix [A(t)] is therefore eliminated by taking the mean value in (Eq.49), while neglecting the harmonics of its Fourier Series components (r=0) [2]. Since [A(t)] is valid only between the entry ( )stφ and exit ( )exφ angles of the cutter (i.e. ( ) 1)j jg φ = , and the process is periodic at pitch

angle ( 2 /p Nφ = π ), the average value of the periodic directional factors becomes:

01 ( )

2

ex

st

xx xy

yx yyp

NA A dφ

φ

α α⎡ ⎤= φ φ =⎡ ⎤ ⎡ ⎤ ⎢ ⎥⎣ ⎦⎣ ⎦ α αφ π ⎢ ⎥⎣ ⎦

∫ (50)

where the integrated directional functions are given as:

1 [cos 2 2 sin 2 ] ,21 [ sin 2 2 cos 2 ]21 [ sin 2 2 cos 2 ]21 [ cos 2 2 sin 2 ]2

exst

exst

exst

exst

xx r r

xy r

yx r

yy r r

K K

K

K

K K

φφ

φφ

φφ

φφ

α = φ − φ + φ

α = − φ − φ + φ

α = − φ + φ + φ

α = − φ − φ − φ

(51)

The time invariant but immersion dependent directional matrix 0[ ]A is substituted in (Eq.48), as opposed to orienting all flexibilities in a fixed direction as approximated by Tlusty [105], Opitz [71] and Weck [117]. Frequency response function matrix ([ ( )])iwΦ identified at the cutter-workpiece contact zone is given as:

( ) ( )( )

( ) ( )xx xy

yx yy

i ii

i i

Φ ω Φ ω⎡ ⎤Φ ω =⎡ ⎤ ⎢ ⎥⎣ ⎦ Φ ω Φ ω⎢ ⎥⎣ ⎦

(52)

The regeneration displacement vector

0 0 ( ) ( ) Tx x y y∆ = − − at chatter frequency cω is a harmonic function expressed as:

( ) [1 ] [ ( )] c ci T i tc ci e e i F− ω ω∆ ω = − Φ ω (53)

where cTω is the phase delay between the vibrations at successive tooth periods T. Substituting ∆(iωc) into the dynamic milling equation (Eq.48), leads to an eigenvalue problem which has the following characteristic equation:

0det[[ ] [ ][ ( )]] 0cI A i+ Λ Φ ω = , (1 )4

ci Tt

N aK e− ωΛ = − −π

(54)

Budak and Altintas [19] considered the harmonics of the directional factor, and formulated the eigenvalue problem which has an increased matrix dimension by 2(2r+1),

det[[ ] [ ][ ( ( )]] 0,

( , ) 0, 1, 2,..,p k c TI A i k

p k r−+ Λ Φ ω + ω =

= ± ± ± (55)

where ωT is the tooth passing frequency and r is the number of harmonics. Although stability (Eq.55) leads to more accurate solution, it is computationally more costly and its accuracy is noticeable only in highly intermittent milling operations where the immersion is very small, i.e. thin web milling and center face milling with a small width of cut [63]. Budak and Altintas instead proposed zero frequency solution (r=0) as shown in (Eq.54). The eigenvalue (Λ) in (Eq.54) can either be solved using iterative techniques as proposed by Minis et al. [66], or a direct solution based on the physics of chatter as proposed by Altintas and Budak [3], who solved the stability as follows. For a given chatter frequency cω , material and tool geometry dependent cutting coefficients (Kt, Kr) radial immersion ( , )st exφ φ and frequency

response function of the structure (Eq.52) are known apriori, which are either measured or predicted from a structural model of the machine. If two orthogonal degrees of freedom in feed (X) and normal (Y) directions are considered and cross coupling is neglected (i.e.

0xy yxφ = φ = ) as considered in the one dimensional model, the characteristic equation becomes just a quadratic function

20 1 1 0a aΛ + Λ + = (56)

where

0

1

( ) ( )( )( ) ( )

xx c yy c xx yy xy yx

xx xx c yy yy c

a i ia i i

= Φ ω Φ ω α α −α α

= α Φ ω +α Φ ω (57)

Then, the eigenvalue ( )Λ is obtained as:

21 1 0

0

1 ( 4 )2 Ra a a ia

Λ = − ± − = Λ + Λ (58)

The solution of the stability is presented in detail by Altintas and Budak in [3], which provides critical depth of cut and spindle speed, i.e. stability lobes, directly as:

2lim

2 (1 )

1 60( 2 )

R

tc

c

aNK

T k n nNT

πΛ ⎫= − + κ ⎪

⎪⎬⎪= ε + π → = =⎪ω ⎭

(59)

where / tanI Rκ = Λ Λ = ψ , and 2ε = π − ψ is the phase shift between the inner and outer waves left on the chip surface. The stability lobes are graphed [2] by varying the number of vibration waves left per tooth period, i.e. k=0, 1, 2... Although only a two dimensional chatter model is presented here, the approach has been extended to three dimensional chatter by simply adding the dynamic flexibility and dynamic cutting forces in the z direction as well as cross transfer functions in all directions if there is strong coupling between them [4]. Face milling with an inclination angle, and milling with circular inserts and flexible spindle axis can be given as examples of the three dimensional version of the milling chatter theory presented here [4],[52]. Altintas’s group and others have experimentally verified the prediction of stability lobes for a significant number of applications ranging from helical end milling to ball end milling, milling with indexed cutters, variable pitch cutters and tapered ball end mills [6][7][8][9][33].

Averaging the time varying constants in direct and cross directions takes the maximum energy in exciting the structural modes, and the coupling between the vibrations in two directions via the cutting process is maintained by Altintas and Budak [3]. The coupling treats the dynamic milling as an eigenvalue problem. Averaging the dynamic resultant force as proposed by Opitz [71], or forcing the resultant force to act at the geometric mean of the cut proposed by Tlusty [105] reduce the eigenvalue problem into a scalar one, hence the stability results can not be the same as given by Altintas and Budak’s [3][19]. Depending on the strength of modes in x and y directions, geometric averaging may shift the energy towards one direction more than the other, hence it may not lead to accurate results when the modes in both directions are equally strong or weak. When the structure is

considerably more rigid in one direction than in the other, two dimensional stability law proposed by Altintas and Budak [3] is reduced to the one dimensional stability law proposed by Tlusty [104]. For example, if

0, 0xx xy yyΦ = Φ = Φ ≠ , the two dimensional chatter theory becomes [19]:

lim2

t yy yya

NK Gπ

=α

(60)

This resembles Tlusty’s theory except that the directional factor is still different than the geometric mean or mean of the resultant force used by Opitz. When all the dynamics are neglected except in one direction, (Eq.59) is reduced to the same formulation proposed by Weck [117] who averaged the directional factors and oriented all the flexibilities in a fixed direction, hence, like Tlusty, reducing the stability to a scalar problem.

5.3 Time domain modeling of dynamic milling The work piece is fed linearly towards a rotating cutter having multiple teeth in milling operations. The dynamic chip thickness given in Eq.(23) considers relative vibrations between the tool and workpiece. Tlusty’s research group pioneered the discrete time domain simulation of dynamic milling [91] [100]. They digitized the dynamic chip thickness at discrete angular intervals d tφ = Ω∆ where ( / )rad sΩ is the angular velocity of the spindle and t∆ is the sampling interval which needs to be selected 8-10 times smaller than any natural mode period in order to capture vibrations. The time domain simulation of milling proposed by Tlusty is adopted by a number of researchers for process planning, evaluation of machine tool dynamics and different cutter geometries [32][118],

as summarized in his book [105]. If the radial width of cut is small and dimensional surface finish needs to be generated during simulation, true kinematics of the dynamic milling is preferred. Altintas’s group developed the true kinematic model of dynamic milling, and included the structural dynamic models of both work piece and cutter at the cutting edge - finish surface contact zones [7][23][67], see Figure 13. For example, a point on the cutting edge has coordinates, which are dependent on spindle speed, feed, tool geometry, radial immersion and depth of cut:

( ), ( ), ( ) , ( ), ( ), ( ), ,= Ω⎡ ⎤⎡ ⎤⎣ ⎦ ⎣ ⎦t t tP x t y t z t f R t x t y t z t (61)

where the cutter axis may vibrate away from the stationary spindle axis in feed (xt (t)) and normal (yt (t)) directions, and z is the elevation of cutting edge point from the tip of the cutter. The cutter has a radius of R and the spindle speed is Ω(t) which is used in calculating the angular immersion of the cutting edge, e.g. ttt )()( Ω=φ . A point on the work piece surface moves linearly towards the rotating but vibrating cutter with a feed speed of f[mm/s], and its coordinates are dependent on the feed speed and part vibrations at that point:

( ), ( ), ( ) ( ), ( ), , ,w w w wP x t y t z t f x t y t z f t=⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ (62)

where xw, yw are the amplitudes of work piece vibrations at this point. The mathematical model of the kinematics, which is implemented in a computer algorithm needs quite a detailed presentation which can be found in [23][67]. The intersection of Pt (t) and Pw(t) gives the cut surface. The instantaneous chip thickness removed by Pt (t) is most correctly evaluated by subtracting the present tool - work piece contact coordinates from the surface generated and tracked by Pw(t) previously. The subtraction is done along the radial vector which passes through the cutting edge point and vibrating cutter center, see Figure 13. The approach allows prediction of true chip load generated by the trochoidal motion of the milling tool and structural vibrations of both the cutter and workpiece. The dimensional surface generation is rather automatic with this method, and the radial and axial tool run-outs are also easily integrated to the model by defining each edge radius from the spindle axis differently. Further details of the complete mathematical model of dynamic milling can be found in [67]. Once the oscillatory chip thickness is evaluated, the dynamic milling forces (Fx(t), Fy(t)) are predicted from Eq.44. The structural vibrations of both the work piece and cutter are predicted by applying cutting forces to each structure at discrete time intervals:

1

1

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

N

x x x xjjN

y y y yjj

m x t c x t k x t F t

m y t c y t k y t F t

=

=

⎫⎪+ + =⎪⎪⎬⎪

+ + = ⎪⎪⎭

∑

∑

&& &

&& &

(63)

where N is the number of teeth on the cutter, m, c, k are the mass, damping and stiffness of either tool or work piece at the contact zone in the feed (x) or normal (y) directions. The dynamic milling equation (Eq.63) is solved numerically by applying Euler, Runga Kutta or Tustin’s approximations.

Figure 13: Surface and chip load evaluation using true kinematics of dynamic milling.

5.4 Experimental and simulation results

Time domain simulations consider true kinematics of the milling, mechanics of cutting, the influence of inner and outer modulation (e.g. exact dynamic chip thickness history), cutter geometry, run-out and the non-linearity such as tool jumping out of cut, and past multiple-regeneration waviness caused by all teeth. Therefore, any frequency domain chatter stability solution is best compared against a numerical, time domain solution. The true kinematics of dynamic milling developed by Altintas’s group in time domain is used to compare various stability models in this article. A sample case in ball end milling of Ti6Al4V Titanium alloy with a two fluted ball end mill is shown in Figure 14, and the details can be found in [8]. The time domain simulation results fully agree with the experiments, as well as with the stability lobes predicted by the analytical model proposed by Altintas et al. [7][8]. The numerical simulation model has been experimentally verified on various cutter geometries such as tapered helical ball end mills, indexed cutters, serrated end mills and ball end mills [6][7][8][9][33][34]. Therefore, it can reliably be used to verify the accuracy of various frequency domain stability lobe prediction models reviewed in this article.

The stability lobe prediction models are compared in slot, half and quarter immersion milling with a four fluted end mill. The cutting coefficients and structural dynamic parameters of the tool are the same for all models and time domain simulations, and are given in Table 1. The feed (x) and normal (y) directions are based on the non-rotating workpiece coordinates. Case 1 - Slot milling (Figure 15): The stability lobes predicted in time domain simulations are compared against the lobes predicted using the frequency domain methods proposed by Tlusty, Opitz’s, and Altintas& Budak. The frequency domain solution proposed by Altintas and Budak closely matches the exact, time domain solution, while Tlusty’s and Opitz’s

approximations over predict the stability. While Tlusty’s model predicts the location of spindle speed pockets with a large depth of cut, Opitz’s prediction shifts them along the speed axis due to dynamics being reoriented or shifted by the average directional factor obtained from

Figure 15: Slot milling simulations

dynamic resultant force. The vibrations in both directions affect the chip load generation and hence the system has coupled dynamics. Two sample time domain solutions are presented at stable (a=20 mm) and unstable (a=30 mm) depths of cut while keeping the spindle speed the same at n=7500 [rev/min]. The Fourier spectrum of the vibrations indicates that the chatter occurs at 721=cω Hz with 0.1 mm amplitude vibrations in y directions. Although the modes are at 510 Hz in x and 802 Hz in y directions, the chatter occurs at 721 Hz due to the influence of directional factors. The stable cut gives low amplitude, constant forces with a deflection of only about 0.040 mm in y direction, which is due to forced vibrations caused by periodic milling forces. The spectrum of vibrations at the stable cut is not visible since the cutting forces are constant without any fluctuation in slot milling when the

Figure 14: Experimental verification of time domain simulation and 2D analytical frequency domain stability

solution for ball end milling of Ti6AL4V alloy [8].

Table 1: Simulation conditions for all cases with 4 fluted cutter with zero helix angle and 19.5 mm

diameter. The feed rate c=0.1mm/tooth is used in time domain simulations of milling cases.

Direction Natural

Frequency

nω (Hz)

Stiffness

]/[106 mNk ×

Damping Ratio ς

X 510 96.15 0.04 Y 802 47.5 0.05

Cutting Constants:

]/[10900 26 mNKt ×= , 3.0=rK . (Al 7050T6)

cutter has four or more even number of teeth. When the teeth have uneven radial run-out, the spectrum would be dominated by the spindle speed frequency. Case 2- Half immersion down milling (Figure 16): All three models predict the chatter stability lobes reasonably well, except that Altintas and Budak’s method

Figure 16: Half Immersion Down Milling Simulations.

predict the lobe shapes similar to the exact solution obtained from the time domain simulations. The effect of the second mode between a spindle speed range of 10000 and 13000 is well predicted by the eigenvalue solution as opposed to less accurate methods. However, considering the simplicity of the scalar solution, Tlusty’s method as well as Opitz’s modification with average orientation factors give a reasonable lobe diagram for practical use in industry. Note that the stability lobes in up milling will be different than in the down milling case, although the width of cut is the same. The difference is totally due to directional factors. It can be noted that the magnitude of the cutting forces and vibrations are almost doubled relative to the beginning of the cut where the chatter has not yet developed. Only the saturation, or the feed rate allowed, limits the exponential growth of the forces and vibrations when the process chatters. 3 – Quarter immersion milling (Figure 17): The three models gave almost identical stability lobes, which all closely agree with the time domain solutions. Both Tlusty’s and Opitz’s models under-predicts the chatter free depth of cut as evidenced by the time domain simulations. The chatter is dominated by the mode in y direction (871Hz) as can be seen by the Fourier spectrum of the vibrations. The directional factor and the damping of the mode move the chatter frequency to a higher value than the dominant natural frequency of the y-axis at 802 Hz. When the immersion is low, the dynamics is dominated by one direction (i.e. y in this case), which reduces the eigenvalue problem to the scalar solution, hence both scalar and eigenvalue models yield to the

same stability solution. However, when the radial immersion is selected in such a way that the structural dynamics in both directions affect the chip regeneration strongly, as in the case of slotting, the dynamics become strongly coupled, and the eigenvalue approach yields more accurate predictions.

Figure 17: Quarter Immersion Down Milling simulations.

Case 4 – Highly intermittent milling with small radial immersion (Figure. 18): The stability of a milling system with a low radial depth of cut is predicted by considering average directional factors [3] and up to three harmonics of tooth passing frequency according to zero and Multi-Frequency solution of Budak and Altintas [19][63], respectively. The cutting conditions and structural parameters are given in Figure 18. While both solutions predicted chatter at point A, zero frequency and multi-frequency solutions predicted contradictory stability results in operating points B and C. The cutting forces and vibrations are predicted using the time domain simulations as shown in Figure 18 a, b, and c. At point A (n = 30000 rpm, a = 2 mm), both zero and multi frequency solutions give chatter, and both solutions, i.e. depths of cut, are almost equal to each other. The force spectrum exhibits dominant tooth passing frequency harmonics plus spread of chatter frequency at the integer multiples of tooth passing frequency. However, there is only one dominant frequency in the spectrum of vibrations in Y direction, which is the chatter frequency. The chatter frequency is obtained as , 947.23c TDSω = Hz,

, 946.9c MFSω = Hz from Time Domain Simulation (TDS) and Multi Frequency Solution (MFS), respectively. This type of regular chatter vibration is named as Hopf Bifurcation in the literature. Point B (n = 34000 rpm, a = 3 mm) is located in the stable, added lobe predicted by the multi-frequency solution, but predicted as an unstable zone by the zero frequency solution. Time domain simulation, presented in Figure 18b, shows that the process is indeed stable. The dominant frequencies of both the tool vibration and

cutting force are only at the tooth passing frequency ( Tω ) and its integer harmonics due to the periodic behavior of the stable milling process.

Point C (n = 38000 rpm, a = 2 mm) is predicted to be stable by the zero frequency solution, but indicated as an unstable zone by the multi frequency solution. The simulated cutting forces and tool vibrations indicate clear instability due to growing amplitudes, and their spectrums are given in Figure 18c. It can be seen that only tooth passing frequency and its harmonics ( Tkω ), as well as half tooth passing frequency and its odd harmonics ( ) ( )( )2 1 / 2Tk + ω are dominant, where k is a positive

integer number. This type of chatter vibration is called Flip Bifurcation in the literature. The multi frequency

solution predicts the most dominant chatter vibration frequency as , 950.5c MFSω = Hz which is almost equal to half of the tooth passing frequency , / 2 950c TDS Tω = ω = Hz. Note that such a spectrum which is dominated at the integer multiples of half tooth passing frequency in cutting forces must not be confused with the forced vibrations, which occur at the integer multiples of tooth passing frequency. The cutting conditions at point C is clearly chatter as proven by the time domain simulations of the process. The mathematical modeling and governing physics of Flip Bifurcation in low immersion milling are given in detail in [19][63]. However, such chatter vibrations and added lobes occur only when the process is severely intermitted with very low radial depth of cut and at high spindle speeds. Most common milling conditions experienced in industry exhibit regular chatter which can be accurately predicted by the zero frequency solution [3]. There has been other stability prediction methods recently presented in the literature by Davies et al. [29], Insberger et al. [48][49], Bayly et al.[13], Kruth et al. [59] and Faassen et al. [36]. It is fair to conclude that the stability of milling has been well understood, and effectively used in industry at moderate to high speeds where the lobes are wide and deep. However, the stability at the low speed range is highly affected by the process damping where the lobes are shallow and narrow, and requires further research. 6 DYNAMICS OF GRINDING The occurrence of vibrations in grinding processes during machining has negative influence on the quality of the workpiece. One has to distinguish between externally-excited and self-excited vibrations [116],[117]. In contrast to externally excited vibrations, the energy for the self-excitation is not introduced from outside, but by the process itself. Similar to metal cutting, the main source of the self-excited chatter vibrations in grinding is also due to the regenerative effect. In this case, the machine structure is excited in the range of its dominant natural frequencies, which are reflected in the form of a waviness on the surface of the workpiece. When the next part of the grinding wheel is cutting in this surface, it leads to a renewed excitation of the machine structure. The process becomes unstable when the damping in the system is insufficient. The stability and monitoring of chatter vibrations in grinding was reviewed in a CIRP key note paper by Inasaki and Karpuschewski [47], and the review of time domain simulation models of grinding is focused in this article. The stability behaviour of plunge grinding and traverse grinding processes with conic roughing zone can be calculated with the aid of limiting phase angle criterion, which is derived from the Nyquist criterion [36][55]. If the phase frequency response of the machine and the limiting phase angle curve have a common point of intersection, a necessary condition for the appearance of regenerative chatter vibrations is fulfilled (Figure 19). If the phase frequency response cuts the first section of the limiting phase angle curve, there is the risk of the workpiece-sided regenerative effect. If the phase frequency response only cuts the second intercept of the limiting phase angle curve, the regenerative effect can be caused on the side of the grinding wheel. Generally, the grinding process can only become unstable, if the frequency response takes values below the phase limit of –90°. Because of the complexity of the contact conditions, an analytic description of the dynamic behavior of grinding processes is difficult to achieve [47].

Figure18.:Stability lobes for Low Immersion Down Milling, Structural Dynamic Parameters in normal (y) direction: 907[Hz],k 1.4e6[m/N], 0.013ny y yω = = ς = ,

and the system is rigid in x direction. The radial depth of cut: 1.256 mm, 3 fluted end mill with 23.6

mm diameter. Feedrate=0.12 mm/rev/tooth. Cutting coefficients (Al-6061): 500tK = 2/ mmN and 0.3rK = .

Time domain simulation methods have been developed to analyze the influence of grinding process parameters and machine compliance on chatter vibrations. In the following, a simulation method to analyze the dynamics of cylindrical traverse grinding processes is described as an exemplary time domain simulation model.

6.1 Dynamic behavior of cylindrical Traverse grinding processes with cylindric roughing zone

Besides a possible generation of waviness on the workpiece and the grinding wheel, the so-called step- wear must be considered in traverse grinding with cylindrical roughing zone. At the beginning of the grinding process, the wear is concentrated to a section of the wheel with the width of the axial feed (Figure 20). Depending on the axial feed, a step-shaped wear profile is formed [110]. This traverse grinding process can be described as a parallel coupling of several plunge-grinding processes in the form of a closed loop system as shown in Figure 21 [1][45][87]. The respective material removal of a specific wear step is influenced by the material removal of the previous wear step. 6.2 Time domain simulation model Based on the closed loop dynamics of the grinding process depicted in Figure 21, a time domain simulation algorithm is developed. The surface of the grinding wheel and the workpiece are represented by fields of equidistant supporting points in the algorithm as shown in Figure 22.

These fields make it possible to calculate the conditions of contact in the form of the intersection of the workpiece and the grinding wheel at any point in time. The dynamic machine compliance can be acquired from measured frequency responses with the aid of modal curve-fitting methods. The simulation model predicts the cutting force, the relative displacement and the surface profile of the workpiece or the grinding wheel at discrete time intervals. The application of the simulation program to the analysis of a dynamic traverse grinding process is described as an example. This grinding process is used in producing large rolls for the paper industry. During the grinding process, distinct chatter vibrations at the 87 Hz appeared. This chatter frequency could be caused by a natural mode at 80 Hz which was visible from the measured frequency response of the machine. Based on the measured compliance behaviour and the corresponding parameters of the force and the wear models, the simulation program was used to calculate the dynamic behaviour of this grinding process. Figure 23 shows the calculated progression of the normal grinding force and the displacement in the grinding gap over the

xw1

xw2

xwn

xw1,i-1

xw2,i-1

xwn,i-1

xw1,i

xw2,i

xwn,i

xGS1

xrel1Q'w1

Q'w2

Q'wn

F'n,1

F'n,n

F'n,2

xrel2

xreln

Fdist.

xGS2

xM

modelof force

contact

geometry ofworkpiece 1

mach. param.

initial surfaceof workpiece

geometry ofworkpiece n-1

resulting surfaceof workpiece

GM

TS

TS

TS

TS

TS

TS

bG'sw1

G'sw2

G'swn

G's1

Tw

G's2

modelof force

Tw

xGSn

modelof force

Tw

G'sn

P1contact

P2

contactPn

mach. param.

mach. param.

Figure 21: Closed loop for cylindrical traverse grinding.

bS

d S

vfa

afa e

d w

afaf

Figure 20: Step-wear during cylindrical traverse grinding.

frequency f0 200 400 600 Hz 1000

0

s

vibr

atio

n am

plitu

de

VW = 0,7 m/s; Vf = 0,7 mm/min•

•

•

VW = 1,175 m/s; Vf = 1,3 mm/min

VW = 1,175 m/s; Vf = 0,7 mm/min

stable process

stable process

instable process 1

2

3

10

xy

z

com

plia

nce

mµ N

52

5

2

100

10-1

10-2

phas

e an

gle ϕ

0°

-90°

-180°

|Gxx| :machine tool compliance

ϕ

limiting phase angle ϕgr

|Gxx|

Figure 19: Stability analysis with limiting phase angle curve.

Figure 22: Conditions of contact.

workpiece length. After a short time, distinct vibration amplitudes increase in the progression of force, as well as in the progression of displacement, which indicate unstable process behaviour. Figure 24 and Figure 25 show the time development of the workpiece surface and the 3rd wear step of the grinding wheel as well as a radial cut of the workpiece and the grinding wheel at the 32nd rotation.

From the waviness of the workpiece, the chatter frequency of the system at f = 90 Hz can be determined with the aid of a FFT analysis (Figure 26). The simulation results show a good correlation with the measured

Based on a technologically optimally designed process, the influence of different machining parameters on the stability behaviour can be analysed and a process optimisation can be performed systematically with the aid of the time domain simulation program. 7 SUPPRESSION AND AVOIDANCE OF CHATTER

VIBRATIONS If not avoided, the chatter vibrations lead to unacceptable surface finish, excessive loads on the cutting tool and spindle causing tool and bearing failures. It is vital to distinguish self-excited, chatter vibrations from the external or forced vibrations during machining, and troubleshoot the source and cause of the problem [54]. The external vibrations can be identified by monitoring the spectrum of vibrations. The forced vibrations occur at the spindle or tooth passing frequencies, but chatter vibrations occur close but not always at the natural frequencies of the machine tool and workpiece system due to orientation of the cut relative to the modes [53]. Both forced and chatter vibrations are best minimized by designing machine tool, fixture and tooling structures which have high dynamic stiffness ( 2kς ), especially in the direction of major cutting forces [56][111]. While both stiffness and damping increases are beneficial, even a reduction of one of these parameters can be helpful if accompanied by an even greater increase of the other one [82]. Usually, the increased stiffness may result in reduced damping, and it may be more beneficial to increase the damping in order to increase the dynamic stiffness of structures [74][92]. Passive damping devices are installed in the flexible machine tool elements, which are along the path of major cutting force directions [83]. Principle passive damping devices, which dissipate the vibration energy, are summarized in Table 2. Active damping devices, which are listed in Table 3, inject vibration energy to the machine tool system in the opposite direction of vibrations produced by the flexible machine tool elements [114]. The active damping devices deliver either force or displacement by computer controlled actuators. The actuators must have sufficient stroke or force delivery capacity with a bandwidth covering major natural modes of the machine tool structure which need to be damped. In addition, the controller design must be robust enough to handle self excited chatter vibrations which exhibit nonlinear transfer functions due to multiple regenerative delay, varying directional factors and cutting coefficients. Once the machine tool and cutting tool are selected, it is still important to avoid chatter vibrations in order to maximize the material removal rate without damaging the machine and workpiece surface, and the available techniques are briefly summarized in the following. The chatter detection and avoidance techniques in grinding were presented by Inasaki et al. [47]. The chatter suppression techniques in metal cutting can be classified

0.7 µm

1,4 µm90°

0°

180°

270°

sectional view(32 rot.)

0 1352l [mm]

πds [

mm

]

1885

0

grinding wheel (simulation, 3.step)

32. rotation

wea

r [µm

]

0

0,5

1,5

1,0

2,0

Figure 26: Simulated grinding wheel surface in transverse grinding.

200

0

Forc

e [N

]

140014000 lw [mm]0

3,0

Dis

plac

emen

t [µm

]

ds = 600 mmbs = 80 mmvs = 45 m/s

ae = 3,75 µmaf = 26,67 mm

dw = 261 mmlw = 1352 mmnw = 48,8 min-1

lw [mm]

Figure 23: Progression of simulated force and displacement in transverse grinding.

90°

0°

180°

270°

1 µm2 µm3 µm

0 1352lw [mm]

π dw [m

m]

820

0

3

2

1

0 dive

rgen

ce [µ

m]

sectional view(32. rot.)

32. rotation

workpiece surface (simulation)

ds = 600 mmbs = 80 mmvs = 45 m/s

ae = 3,75 µmaf = 26,67 mm

dw = 261 mmlw = 1352 mmnw = 48,8 min-1

Figure 24: Simulated workpiece surface in transverse grinding.

(measured)

0 2000

160

Am

pl. [

dB]

f [Hz]

FFT

f [Hz]

[m/s2]

t [s]

0

0,1

0

80

0 200

87 Hz 90 Hz

acceleration(simulation) FFT analysis

Figure 25: Measured and simulated frequency spectrum in transverse grinding process with

chatter.

under off-line during process planning or on-line, sensor assisted techniques as presented in the following.

Table 2: Passive vibration damping principles (source: M. Weck, WZL, TH Achen).

7.1 Process planning with lobes The chatter vibrations are best avoided by selecting stable cutting conditions predicted from the stability lobes. For example, the stability lobes shown in Figure 17 indicate that the best cutting condition in quarter immersion milling is at 12,000 rev/min with a depth of cut of 60mm in milling Al7050 alloy on a particular machine with the dynamics given in Table 1. When compared to cutting condition at 7000 rev/min and maximum depth of cut of 20mm, the productivity can be increased by 500% at 12,000 rev/min where the stability lobe is deepest and widest. The physical mechanism behind the stability lobe is illustrated in Figure 27 with the following example. If we

consider a milling operation which chatters at frequency fc[Hz], while cutting at a spindle speed of n[rev / min].

Table 3 : Active Vibration Damping Principles (Source : M. Weck, WZL, TH Aachen.)

Grinding spindle bearing controlled by a Piezo Actuator.

Cylindrical grinding machine tailstock controlled by a Piezo Actuator.

Hydrostatic milling spindle bearing controlled with a hydraulic actuator.

With a cutter having N teeth. The tooth passing frequency is ft[Hz] = n/(60N) and the tooth passing period is T[sec] = 1/ft = 60N/n. The total angular travel of the vibration

Active Dampers: Tables, rams, cross-beams and flexible workpieces.

Mechanical Impedance control: Lathe tool holders.

Friction Damper: Main machine frames.

Auxiliary mass damper: Machine columns, rams, tables, spindles and boring bars.

Tuned damper: Milling spindles, rams, and boring bars.

Squeeze film dampers: Machine tool spindles.

Hydrostatic damping of milling and grinding spindles.

wave within one tooth period is expressed from (Eq.59) as:

= = + =2 2 , 2 ( / )c c c tT f T k frac f fω π π ε ε π (64)

If the tooth passing frequency is selected as an integer fraction of chatter frequency, i.e.

/ 1,2,.., ( / ) 0c t c tf f k frac f fε= → = = , the regenerative time

delay term (ε ) in dynamic chip thickness expressions (Eqs.5, 33, 53) can be cancelled as:

2 /( ) (1 ) ( )02( ) (1 ) ( )0 00

c t

t

j f fh s h e y sdj kh s h e e y s hd 144424443

− π ⎫= − − ⎪⎪− π −ε ⎬= − − = ⎪

⎪⎭

(65)

When the tooth passing and chatter frequencies match, exactly one vibration wave is left at each tooth period, which corresponds, to the lobe number one (i.e. k=1). Hence highest depth of cut is possible at this high speed. The peaks of the successive lobes correspond to integer divisions of chatter frequency, and produce integer numbers of full vibration waves at each tooth period. The lowest axial depth of cut corresponds to having worst regenerative phase shift (-180 degree) that leaves a half vibration wave at the end of each tooth period. Due to largest regeneration at this speed, where the inner and outer waves have opposite phase, the dynamic chip thickness grows and becomes largest during chatter. Hence, when the tooth passing frequency, which is equal to number of teeth times spindle speed, is selected as an integer fraction of chatter frequency, the regeneration tends to be minimum and maximum stability is achieved. Although the machine is excited at its natural frequency at this condition, the chip thickness remains the same as programmed static value, see (Eq.65). Most chatter suppression techniques rely on this fundamental phenomenon of minimizing regenerative phase delay. However, such a condition is possible only by running the machine at speeds close to its natural frequencies, which lead to forced vibrations with increased amplitudes due to resonance. Spindle speeds, which are shifted slightly from the peak of the lobes, can be selected to avoid large resonance vibrations.

The stability lobes become smaller at low speeds. A slight change in the low speed may easily create a fraction of a vibration wave, i.e. large regeneration phase, which pushes the system to instability again. The stability lobes are widest at high spindle speeds, which are close to the natural modes of the machine tool and workpiece in metal cutting operations. However, the spindle must

have sufficient torque and power, and the cutting tool grade must be able to withstand against high thermal loads, which increase with the cutting speed. Typical binding materials in carbide and CBN tools can withstand temperatures up to 1000 and 1300 Celsius, respectively. Materials with low yield strength like Aluminium alloys melt before the rake face-chip contact temperature reaches to tool material’s diffusion wear limits. On the other hand, thermal resistant alloys such as Titanium and Nickel based metals, need to be cut at cutting speeds under 100 m/min which usually corresponds to the low spindle speeds where the lobes are very narrow and impractical to rely on. Variable pitch cutters disturb the regeneration mechanism in milling, and can be effectively used in low speed applications at the expense of irregular chip load distribution among the teeth, which lead to non-uniform tool life [9][21][22][109]. In summary, the predictions of stability lobes complemented by the time domain simulation of metal cutting and grinding to select most optimal machining conditions that do not violate the machine tool, workpiece and cutting tool constraints, are recommended in practice. The machine tool engineers can use the stability lobe and process simulation tools to design spindles, columns, tables, guides and feed drives which can operate at targeted machining operations without chatter. Various damping mechanisms can be incorporated to the machine tool structures by using the chatter stability laws summarized in the article for metal cutting and grinding operations.

7.2 Real time chatter vibration avoidance techniques There have been significant efforts in adaptive control of chatter vibrations using actuators as summarized under the principles of active dampers in Table 3. Their objective is to increase the dynamic stiffness and minimize the regenerative phase shift. However, most of the techniques were not able to migrate from the research laboratories to shop floors due to two reasons. Chatter vibrations occur at the low frequency modes of the structure, which can be up to 100 Hz in low speed machining and grinding, or above 400Hz at high speed machining operations where the spindle and tool holder modes become more dominant. The low frequency modes originate from machine tool components with large masses, such as column and table. The actuator must have a large dynamic force capacity with at least 5-10 micrometer stroke delivery with 100 Hz bandwidth. Active chatter vibration control at high speed machining requires a bandwidth higher than 1000 Hz from the actuators having a 10-20 micrometer displacement stroke in order to cover spindle and workpiece modes. Unfortunately, the existing actuators cannot meet such bandwidth and dynamic displacement requirements. Furthermore, the closed loop dynamics of the chatter system has a regenerative feedback delay, which makes the system nonlinear and difficult to control in real time. The laboratory trials gave promising results only when the control law and piezo actuator are dedicated to fixed, flexible structures such as a boring bar [97]. The alternative to real time control of chatter is to avoid it by monitoring the process using vibration, force or acoustic sensors [26][47][77]. Weck demonstrated the detection of chatter vibrations in milling by filtering the tooth passing frequency from the power spectrum of cutting forces [112]. He searched the stability pocket by matching the tooth passing frequency with the chatter frequency [113]. Weck also proposed automatic distribution of depth of cut in turning until the chatter diminished. Delio, Smith and Tlusty developed a real time chatter avoidance system by detecting the frequency and presence of chatter using sound spectrum [31]. They automatically matched the tooth passing frequency to the

Figure 27: The stability pockets correspond to having integer number of waves between the tooth

periods in milling. [Source : UBC M.A.L.]

integer sub-harmonics of the chatter frequency either directly within CNC or by recommendation to the operator using a portable computer system [90]. The identification of stability lobe pockets requires measurement of vibration frequency using sound, accelerometer or force sensors. The tooth passing frequency and its harmonics represent the forced vibration, and could be filtered away before detecting the presence of chatter frequency. Since the first lobe corresponds to leaving one integer number of chatter vibration wave in one tooth period, the first spindle speed can be tried by matching the tooth passing frequency to the measured chatter frequency. If the depth of cut limit resides in the lobe, the regenerative phase delay which causes the chatter becomes minimum, and the chatter is suppressed. If the spindle does not have such a high speed, the integer divisions of chatter frequency can be selected as tooth passing frequency which will correspond to the 2nd, 3rd .., , kth stability lobes. The approach may not lead to guaranteed stability if the selected depth of cut is more than the actual stability lobe limit. However, the stability lobe prediction requires the measurement of Frequency Response Function of the machine tool-workpiece system, cutting coefficients and the execution of stability law. The in process detection and lobe search algorithms proposed by Weck et al. [112] and Delio et al. [31] allow a practical chance of suppressing chatter during the machining operation. Mitsuishi et al. [68] detected chatter using force sensors installed on each axis of the machine, and calibrated stability lobes during milling. The stability lobes were connected to a CAD system to modify the NC tool path. However, such techniques require real time modification of the NC program that is not trivial in milling complex parts such as dies and molds. Sexton and Stone [86] proposed variation of spindle speed to disturb the regeneration of chatter waves in turning. De Canniere et al. [30], Jemielniak et al. [51] and Altintas et al. [5] tried the same approach in turning and milling, respectively. Sastry et al. [84] analyzed the torque and power requirements from a spindle as well as the chatter stability for the effective application of variable spindle speed to suppress chatter vibrations in milling. The variable spindle speed becomes effective only when the chatter vibration frequency is low, and the spindle drive has sufficient bandwidth and power to influence the regenerative phase shift in successive tooth periods in milling, or in successive spindle rotations in turning. 8 CURRENT RESEARCH TASKS IN DYNAMIC

CUTTING AND GRINDING The prediction and avoidance of chatter vibrations have shown significant progress and found industrial use during the recent decade. The most successful applications have been demonstrated in high-speed milling where the stability lobes are wide allowing large depth of cuts. The time domain simulation of milling and grinding, as well as frequency domain prediction of stability in milling have been widely studied with successful results. The linear chatter stability laws expressed in Equations (7), (59), (65) or in the feedback block diagram shown in Figure 1 assume that the system stability is independent of input variable feed-rate, and the dynamic cutting force is linearly proportional to the axial depth of cut. This assumption is valid for tools with zero nose radius, or when the axial depth of cut is significantly larger than the tool nose radius [43]. However, cutting tools used in turning and boring operations usually have a nose radius and approach angle, which lead to irregular distribution of uncut chip area as a nonlinear function of feed-rate and radial depth of cut, see Figure 5 [61]. The chatter

vibrations usually occur in the radial and feed directions, which lead to irregular distribution of chip thickness along the cutting edge in turning and boring operations. As a result, the cutting force coefficients become dependent on the changes in vibrations and the feed rate, which make the process stability too nonlinear to solve in frequency domain. There have been some attempts to linearize the process around small variations in vibrations and feed-rate, but with limited success due to the presence of severe nonlinearity in cutting force coefficients. Lazoglu et al. [61] developed a time domain simulation model for dynamic turning and boring operations. They considered an exact kinematic model of the process where the workpiece rotates, and the tool is linearly fed while vibrating in radial and feed directions. The computational time of the numerical simulation of turning and boring became unreasonably long. In addition, the structural dynamics of machines may change as a function of tool position or spindle speed, which makes the FRF of the machine nonlinear [88]. The chatter stability is best studied by stability solutions, which can handle severe nonlinearities in the cutting force coefficients as well as in machine tool dynamics. Such methods are presently not available in the literature and are open to research efforts. There are significant number of metal cutting operations which are affected by chatter vibrations, but are yet to be studied extensively. They can be listed as circular sawing [65][124][125], broaching, gear shaping, tapping, circular-helical milling, mill-turning, parallel machining on a single part by multiple spindles, combined drilling-boring-milling operations using a single cutter. The research challenges in grinding are mainly due to nonlinear behavior of the process. The contact area between the grinding wheel and the workpiece, the variation of grinding coefficients as a function of grinding wheel condition and contact zone make the frequency domain solutions difficult to implement in practice. Although the present actuators do not have the desired bandwidth and power to adaptively control chatter vibrations in real time, research in modeling the machining process dynamics and feedback control are still important topics to be studied. Perhaps, the first steps would be to focus on the increase of dynamic stiffness of spindle shafts, bearings, boring bars, work-holding devices and milling heads using active damping devices before attempting to cover a wide range of applications simultaneously with a general solution. 9 CONCLUSION The paper summarizes the state of research knowledge in dynamics of metal cutting and grinding operations. Since Tlusty [104] and Tobias [106] proposed the pioneering orthogonal chatter stability laws in 1950s, significant progress has been made in modeling machining processes. The frequency domain chatter stability laws in high speed milling have been well developed and used in industry effectively. Significant research efforts have been reported in time domain modeling of turning, boring, milling and grinding operations which allow the analysis of dynamic machining when the machine tool and workpiece stiffness, work material, tool geometry and cutting conditions are varied. The frequency and time domain simulation models are effective tools to improve machine tool design and for optimal planning of machining operations. There are still unsolved research challenges in dynamic cutting and grinding. The solution of chatter stability for tapping, gear shaping, broaching and parallel machining have yet to be developed for practical use in industry.

The chatter stability is still not solved when the process is highly nonlinear due to time varying and nonlinear cutting coefficients including the process damping at low speeds, and when the structural dynamics of the parts and machines vary along the tool path. It is most desirable to integrate process simulation and chatter stability algorithms directly to CAD/CAM systems so that the machining operations can be optimally planned and tried in a virtual environment before testing them on costly trial cuts on the shop floor. ACKNOWLEDGEMENT The metal cutting dynamics part of the paper was written by Professor Y. Altintas. His assistants J. Roukema, G. Suren , D. Merdol and N. Kardes provided the figures in drilling, turning and milling, respectively. Prof. P. Vanherck (PMA, K.U.L) provided feedback on CIRP task force study on dynamic cutting force coefficients [102]. Prof. P. Bayly checked the drilling section. Prof. K. Weinert contributed the deep hole drilling section. Dr. M. Zatarain from Tekniker checked the dynamic milling section. Prof. Weck prepared the grinding section of the paper and provided the damping tables with assistance from A. Schulz at WZL. A number of CIRP colleagues sent their relevant articles, which assisted in the preparation of the article. References [1] Alldieck, J., 1994, Simulation des dynamischen

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