Z Dombovari

A Iglesias

G Stepan

Study of the Tuning of Variable Pitch Milling Cutters

Keywords: stability, milling, tool geometry, variable pitch

It is well known that variable pitch cutters may improve stability of a given milling operation. Several methods are known for tuning the consecutive pitch angles to approach idealistic phase shift on regeneration. Most of the tuning procedures are based on the measurement of the chatter frequency. However, these design procedures completely disregard the effect of the harmonics of the milling process. In this study, the efficiency of these methods is verified in milling for an interrupted cutting engagement. The critical analysis of different tuning methodologies are carried out by means of accurate stability charts constructed with semi-discretization. It is shown that the effectiveness of tuning can be improved further with numerical optimization techniques.

Introduction

The dynamic behavior of regenerative cutting processes can be influenced through the generated cutting force. Apart from the material properties, it is the edge geometry that affects the most the relative magnitude and the regeneration of the instantaneous cutting force. In this manner, any attempt to distort both the strength and the length of the regenerative delays most likely leads to better conditions for stable stationary cutting [START_REF] Munoa | Chatter suppression techniques in metal cutting[END_REF]. As impacts are reduced by introducing helices, regenerative delays can be distorted by irregular spacing of edge portions in the direction of the cutting speed, or by the variation of the spindle speed [START_REF] Zatarain | Stability of milling processes with continuous spindle speed variation: Analysis in the frequency and time domains, and experimental correlation[END_REF]. Discrete variation of regenerative delays can be introduced by simple uneven pitch angles or by irregularities in local subsequent radii causing missed cuts e.g. in case of serrated cutters [START_REF] Dombovari | The effect of serration on mechanics and stability of milling cutters[END_REF][START_REF] Dombovari | General Milling Stability Model for Cylindrical Tools[END_REF].

While continuous variation on the delay's weights can be introduced by helix angle variations [START_REF] Otto | Stability of milling with non-uniform pitch and variable helix Tools the frequency and time domains, and experimental correlation[END_REF].

Due to the uneven pitch geometry, the corresponding milling processes are subject to the effect of the irregular interaction of the time-periodicity and the distorted constant delays. This special geometry changes the regenerative phases between the past and the present vibration pattern in modulo, which effect was historically modelled by unevenly displaced multi-edge orthogonal planing operation by Slavicek [START_REF] Slavicek | The Effect of Irregular Tooth Pitch on Stability of Milling[END_REF]. In this work significant gain was achieved in depth of cut compared to conventional multi-edge-planing. Average directional factors were used by Opitz et al [START_REF] Opitz | Improvement of the Dynamic Stability of the Milling Process by Irregular Tooth Pitch[END_REF] to model the experimentally observed increase on stability by considering the simplest uneven pitch case with two teeth. In terms of rectilinear tool motions, computer simulations were carried out by Vanherck [START_REF] Vanherck | Increasing Milling Machine Productivity by Use of Cutters with Non-Constant Cutting Edge Pitch[END_REF] for different set of varying pitch angles showing the behavior of distinct non-conventional tool geometries. Also, applying numerical simulations, stability properties of variable pitch cutters were simulated in the work of Tlusty et al [START_REF] Tlusty | Milling Cutters With Irregular Pitch[END_REF]. Significant changes in stability were shown in [START_REF] Tlusty | Use of Special Milling Cutters Against Chatter[END_REF] without mentioning geometrical design strategies: improved mechanical model was used by fully respecting the varying directional orientations of the variable pitch geometry.

Single frequency variable pitch model was presented by Altintas et al [START_REF] Altintas | Analytical Prediction of Chatter Stability and Design for Variable Pitch Cutters[END_REF] where the eigenvalue problem is formulated and the stability behavior is shown with respect to the pitch angles introducing real case dynamics. Based on time averaged single frequency solution Budak [START_REF] Budak | Improving Productivity and Part Quality in Milling of Titanium Based Impellers by Chatter Suppression and Force Control[END_REF][START_REF] Budak | An analytical design method for milling cutters with nonconstant pitch to increase stability, part I: theory[END_REF][START_REF] Budak | An analytical design method for milling cutters with nonconstant pitch to increase stability, part II: theory[END_REF] introduced an analytical design methodology based on the phase differences between consecutive constant delays, the effectiveness of which was confirmed with in industrial application. In the work of Olgac and Sipahi [START_REF] Olgac | Dynamics and Stability of Variable-pitch Milling Dynamics and Stability of Variablepitch Milling[END_REF], a unique scheme, the socalled cluster treatment of characteristic roots paradigm was used to investigate the effect of two constant delays on the time-averaged dynamics of milling operation with variable pitch cutters. Time finite element method and time averaged semi-discretization were used by Sims et al [START_REF] Sims | Analytical Prediction of Chatter Stability for Variable Pitch and Variable Helix Milling Tools[END_REF] to build general models of variable pitch and helix tools, around which optimization process was established in [START_REF] Yusoff | Optimisation of variable helix end millings tools by minimising self excited vibration[END_REF][START_REF] Yusoff | Optimisation of Variable Helix Tool Geometry for Regenerative Chatter Mitigation[END_REF] based on genetic algorithms.

The inconsistency of the time period and the regenerative delays in case of variable pitch cutters results in that higher harmonics can have significant strength in the regenerative force even in the case of noninterrupted cutting operations, causing discrepancies in time averaging methods showed in Sellmeier and Denkena [START_REF] Sellmeier | Stable islands in the stability chart of milling processes due to unequal tooth pitch[END_REF]. The importance of the optimization of variable pitch was pointed out in the experimental work of [START_REF] Stepan | Cylindrical milling tools: Comparative real case study for process stability[END_REF]. The construction of transition matrix and the unavoidable eigenvalue calculation were joined by subspace iteration in order to decrease the overall order of the numerical methods (see [START_REF] Zatarain | Stability analysis of milling with irregular pitch tools by the implicit subspace iteration method[END_REF]).

Similar effect can also be reached in drilling and in turning processes, when multiple regenerative delays are introduced instead of the conventional single one. In case of turning, optimized multi edge cutting can disturb the regenerative phase between consecutive edges as presented in Budak and Ozturk [START_REF] Budak | Dynamics and stability of parallel turning operations[END_REF].

Model Description

For simplicity, helix is not considered in this study which can practically correspond to inserted milling with slight helix angle. In this case the regenerative delays τ i can be calculated with the help of the period of the tool T = 2 π/Ω and the pitch angles φ p,i between consecutive edges as

  i i , p   , (2.1)
where the spindle speed is Ω (rad/s) = 2 π n (rpm) / (60 s/min). Due to the successive regenerative waves, phase can be defined absolutely as ε i and in modulo as ϑ i in the following way

i i i l     π 2 :
(2.2) for each (i mod Z) and (i+1 mod Z) pair of teeth. The integers l i = 0, 1, … are the number of whole waves copied in the surface originated from the main chatter angular frequency ω of the self-excitation. Thus, the different properties of the regenerative phases can be expressed for uneven pitch cutters as (round down is )

i i     ,   π 2 i i l    and π 2 mod i i     . (2.3)
Unlike regular cutters, the principle period T p of the milling process is not the tooth passing period T Z = T/Z. Depending on the actual configuration, it is determined by the natural number divisor

  Z l k Z l k Z N 1 , mod 1) ( p, rank /      .
(2.4)

as T p = T / N.
In this framework, we assume momentary chip thickness variation between two consecutive edges as

) ( cos )) ( ) ( ( ) ( sin )) ( ) ( ( )) ( ( )) ( ( ) ( t t y t y t t x t x f t t t h i i i i i i i i i                x f n T (2.5)
for a milling operation with feed

f i = f φ p,i /(2 π) in x direction.
Thus,

f i = col(f i , 0, 0) and n ( i (t)) = col(sin  i (t), cos  i (t), 0). The feed per revolution is f (m/rev), while the position angle       1 1 , p ) ( i k i i t t   (2.6)
determines the position of the ith tooth geometrically. Assuming (for simplicity) linear cutting force characteristics with edge coefficients K e and cutting coefficients K c = col(1, K c,r , K c,a ) (see Table 2.1.), the regenerative cutting force has the form:

)), ( ) ( ( ) ( ) ( )) ( ( )) ( ( )) ( ( )) ( ), ( , ( 1 , c c c, e 1 i Z i i t i t i Z i i k t t t K a t t h K t t g a t t t                 x x A G K K T x x F (2.7)
for lead angle  = 90 (deg), where the transformation matrix T and the screen function g is defined in [START_REF] Dombovari | The effect of serration on mechanics and stability of milling cutters[END_REF], and k = 1, 2, …, N  . The regenerative cutting force in (2.7) depends on the present x(t) and the delayed x(t k ) positions of the tool where x(t) = col (x(t), y(t), z(t)). Furthermore, it can be separated into periodic state independent part G(t) = G( t + T p ), and linear state dependent part with periodic coefficient matrices [START_REF] Iglesias | Analytical expressions for chatter analysis in milling operations with one dominant mode[END_REF].

A i (t) = A i ( t + T p )

Milling Dynamics

By using the same procedure described in [START_REF] Budak | An analytical design method for milling cutters with nonconstant pitch to increase stability, part I: theory[END_REF][START_REF] Iglesias | Analytical expressions for chatter analysis in milling operations with one dominant mode[END_REF], the determination of asymptotic stability of the periodic stationary solution with Ω p = 2 π / T p leads to the following characteristic equation

0 ) e ( ) ( det 1 j c,               Z i i i t i K a D E I I     . (2.

8)

Considering h harmonics in (2.8), we have

) ( diag ) ( p      l h h l   H  , i l h h l i , p j e diag      I E , h h k l i k l i     , , ] [A D and        l t l i l i t p j , e ) ( A A . (2.9)
Basically, (2.8) is a truncated Hill's representation by using the frequency response function (FRF) H(ω) at the tool tip [START_REF] Mancisidor | Fixed Boundaries Receptance Coupling Substructure Analysis for Tool Point Dynamics Prediction[END_REF], which describes the dynamics of the milling process. This solution is referred to in the literature as multi-frequency solution [START_REF] Budak | Analytical Prediction of Chatter Stability in Milling -Part I: General Formulation[END_REF][START_REF] Merdol | Multi Frequency Solution of Chatter Stability for Low Immersion Milling[END_REF][START_REF] Munoa | Interaction between multiple modes in milling processes[END_REF].

In order to deal with the analytical literary cases, the single frequency solution is introduced here for

0 ) e 1 ( ) ( det 1 0 j c,                   N k t k K a A H I . (2.10)
corresponding to the model in [START_REF] Budak | An analytical design method for milling cutters with nonconstant pitch to increase stability, part I: theory[END_REF], where the harmonics

     Z i i i k N 1 , 0 , c 0 d ) ( ) ( π 2 ex en A n K T A       T , (2.11) while  k,i =1 if  k =  i , otherwise  k,i = 0 with k = 1, 2, …, N  .
The radial immersion is simply given by the entry  en and exit  ex angles (see Fig. 2.1.a).

Generally, the solution of (2.8) and (2.10) can be calculated by solving the eigenvalue problem for the depth of cut a [START_REF] Budak | An analytical design method for milling cutters with nonconstant pitch to increase stability, part I: theory[END_REF] or by using the bisection method [START_REF] Bachrathy | Improved Prediction of Stability Lobes with Extended Multi Frequency Solution[END_REF] in the parameter space (a, Ω, ω).

Simplified Case

In this work, we deal with the differences between the historical analytical tuning procedures and the possibly best achievable one. Considering that every new pitch angle increases the dimension of the optimization parameter space by one, a simple and practical oneparameter case (with φ p,1 ) can be introduced with the symmetric arrangement of Z = 4 flutes with κ = 90 deg lead angle (see Fig. 2.1.a). In this case the following notation can be introduced φ p,i = φ p,1 , π  φ p,1 , φ p,1 , π  φ p,1 with N = 2 in (2.4). Also the notation can be introduced describing the angle differences as φ p,2 = φ p,1 +Δφ p,2 . As further simplification, only one spatial direction is considered in the x direction, which makes the matrices at (2.9) scalars along with the considered single mode (see

Table 2.1.) in (2.10), resulting in 0 ) e 1 ( ) ( 1 1 0 j c,          N k t A H K a k , (2.12)
The milling process is chosen to be interrupted to show the discrepancies on analytical tuning procedures that are all based to some extent in the single frequency solution (2.10). Moreover, this interrupted case is more practical process-wise, since it models finishing operation for which vibration needs to be attenuated to achieve good surface quality. Further details of the considered case can be followed in Table 2.1.

Different Tuning Schemas

The three main literary analytical tuning techniques are discussed in this section in chronological order. We mention these solutions as Slaviček's [6], Engin's [START_REF] Altintas | Analytical Prediction of Chatter Stability and Design for Variable Pitch Cutters[END_REF] and Budak's [START_REF] Budak | An analytical design method for milling cutters with nonconstant pitch to increase stability, part I: theory[END_REF] tuning methods referring to the main contributors' names. Although strictly speaking, Slaviček's dealt with planing only in his work, his tuning methodology can easily be applied for milling. All methodologies are simplified to the case introduced in Section 2.

Slaviček's tuning

Originally, this method is a graphical way of tuning based on the different phase shifts in the scalar single frequency formulation introduced in (2.12). Substituting H(ω) = r (ω)e j ψ(ω) , considering the regenerative phases at (2.3), and keeping in mind that the geometric arrangement introduced in Fig. 2 Contrary to the original work, the force does not introduce additional phase shift represented by ρ in [START_REF] Slavicek | The Effect of Irregular Tooth Pitch on Stability of Milling[END_REF]. By using the intricate trigonometrical manipulations in [START_REF] Slavicek | The Effect of Irregular Tooth Pitch on Stability of Milling[END_REF], (3.1) can be rearranged as

    j 0 c, j j e 1 ) cos e 4 ( : e       A r K a N t , (3.2)
where the average regenerative phase is ε = (ε 1 + ε 2 )/2, and the phase difference is Δ = (ε 2  ε 1 )/2 = Δε 2 . The resultant phase shift δ in (3.2) has the least chance (see [START_REF] Slavicek | The Effect of Irregular Tooth Pitch on Stability of Milling[END_REF]) to "intersect" with ψ if δ varies in the slightest way, that is cos Δ  0. Taking into account (2.1) and (2.3), the tuning criterion is given as

2 π n    π 2 π 2 2 , p k          , (3.3)
where historically n = 1, 3, 5, … and k = 1, 2, 3, … . That is, knowing the main vibration (chatter) frequency ω and the spindle speed Ω the pitch difference Δφ p,2 can be designed setting k considering geometric design restrictions (φ p,min , φ p,max and Δφ p,min ). Bearing in mind (3.2) and (3.3), the methodology for this simple one dimensional, single frequency and simple pitch arrangement case (Fig. 2.1.a) can be generalized for multiple modes, too.

Engin's tuning

The tuning methodology presented in [START_REF] Altintas | Analytical Prediction of Chatter Stability and Design for Variable Pitch Cutters[END_REF] is based on an initial stability calculation (e.g. with single frequency or semi-discretization) where the regenerative phase is "picked up" from the diagram to ensure the positioning of the stability pockets (sweet spots or stability resonances, e.g. n = 2670 rpm in Fig. 3.1.d). A stability pocket actually corresponds to two different regenerative phase shifts. In the simple case presented in Fig. 3.1.c), these are actually ϑ sp =180 deg or 360 deg. According to [START_REF] Altintas | Analytical Prediction of Chatter Stability and Design for Variable Pitch Cutters[END_REF] the one which changes less with the spindle speed ensures more robust solution.

In this manner the design criterion is to have the regenerative phase shift (in modulo) for the stability pockets ensured by the first pitch angle

π 2 sp 1 , p k       and k = 1, 2 , 3, … . (3.4)
This design criterion is taking the complicated dynamics (even multi-dimensional one) into account by an initial calculation, predicting the necessary regenerative phase in modulo ϑ sp . The integer k has to be set to satisfy geometrical design criteria (see previous subsection) and to have φ p,1 as close to 2π/Z as it is possible.

Budak's tuning

This tuning procedure is based on the eigenvalue consideration of single frequency solution at (2.10). This results in parametric solution for the depth of cut a as explained in [START_REF] Budak | An analytical design method for milling cutters with nonconstant pitch to increase stability, part I: theory[END_REF]. Considering 3 spatial dimensions (xyz) in (2.10) the following holds

) ( ) ( π 2 ) ( I , c    S K a t    , ), ( j ) ( ) ( I R         (3.5)
where

          N k t k K a 1 j , c ) e 1 ( π 2 ) ( and        N k k S 1 sin ) ( .
The basic idea here is that, the limit depth of cut a at (3.5) is maximal if S is minimal or in extreme cases 0. Applying geometry in Fig. 2

) ( sin sin ) ( 2 2 2 1 1                   S (3.6)
In this case, this methodology gives the same criterion as Slaviček's tuning at (3.3).

The tuning is general since the eigenvalue solution of (2.10) can be made for multi-dimension and multi modal cases, too. However, the S(ω) = 0 would be true only in limiting case which might interfere with Λ I (ω).

Comparison of analytical methods

In this simplified one-parameter optimization case Slaviček's and Budak's tunings are identical. The comparison of these can be followed in Fig. 3.1. where the first pitch angle was designed to be smaller and closer to 2π/Z. This means Engin's and the other two tuning methodologies introduce opposite trends in φ p,1 seemingly resulting in identical tuning for the minimum in this one DOF single frequency case. Also toward the larger lobe number the modulo of regenerative phase eventuates smaller deviation for Δφ p,2 (see Fig. 3.1.b). This makes variable pitch milling tools extremely difficult to accurately tune and produce in low spindle speed zone.

Applying semi-discretization

In order to have the accurate stability chart even for the described interrupted case (Table 2.1.), semidiscretization (SD) [START_REF] Insperger | Semi-discretization method for delayed systems[END_REF] method is applied. By applying SD, the prediction of the main vibration frequency is not trivial at all. This is because the critical multiplier μ c (closest to magnitude 1) refers directly only to the smallest possible modulation (base frequency, ω b ) of the main vibration frequency ω described in [START_REF] Dombovari | Prediction of multiple dominant chatter frequencies in milling processes[END_REF], that is The main vibration frequency can be determined by performing low resolution fast Fourier transform (FFT) on the periodic part of the critical eigenvector of the transition matrix [START_REF] Dombovari | Prediction of multiple dominant chatter frequencies in milling processes[END_REF]. Thus, it can be set by the largest modulation number q on the low resolution FFT as

| | p b c,    q   . (4.2)
In this way, it is possible to apply tuning criteria (3.3), (3.4) and (3.6) by using a preliminary calculation determining the corresponding main vibration (chatter) frequency ω. Also it is worth to mention that after the tuning, the ω can slightly change, but in this one DOF case this effect is negligible.

A brute-force iterative (BFI) method is also implemented, which only relies on the SD method. The optimum is achieved by applying basically bisection scanning on φ p,1 minimizing the magnitude of the corresponding critical multiplier | μ c |. Afterwards this method is referred as "BFI tuning".

Applying for one specific tuning

By using the SD method, the quality of the tuning can be rated by calculating the accurate stability behavior of the introduced interrupted milling process (Table 2.1.). In Fig. 4.1., the different tuning techniques are compared away from the single frequency minimum of the first lobe at n = 1800 rpm. In this case, Budak's and Slaviček's tuning (afterwards "BS-tuning") serve different pitch arrangement than Engin's tuning (afterwards "E-tuning"). It can be followed in Fig. 4.1. that E-tuning completely fails for this given case, while BS-tuning really improves the stability properties, in spite of the fact that it is "chopped off" by the now more intricate flip instability corresponding to the principle period T p . The best tuning is given by the BFI tuning that eliminates the effect of the flip instability in this case. The actual tuning results are listed in Table 4.1. It can be recognized immediately the E tuning methodology does not serve good geometrical arrangement in pitch spaces to gain stability. The BS tuning results in better optimized geometry, however, the BFI tuning induces even higher stability limits (see Fig. 4

.2.).

Similarly to Fig. 3.1. opposite trend between BS and E tunings are appearing. Not surprisingly, the brute force method (BFI tuning) follows different optimized geometric arrangement ensuring the best possible tuning.

Conclusions

In this paper, a simple variable pitch tool arrangement was investigated. It was shown, that the single-frequency based analytical solutions of the literature are not efficient enough to find optimal tuning. Based on the semidiscretization method, a brute force methodology was developed that is able to determine the best tuning for each parameter configuration by minimizing the critical characteristic multiplier of the corresponding timeperiodic delayed system. The simple practical case-study showed that there is a potential in developing improved tuning algorithms for variable pitch cutters.

Fig. 2 .

 2 Fig. 2.1. a) simplified symmetric teeth arrangement, b) phase shifts

  .1.a) with (N  = 2), the following expression is obtained:

Fig. 3 .

 3 Fig. 3.1. a) tuned first pitch angle, b) designed pitch differences, c) regenerative phase and d) ideal single frequency solution for the process in Table 2.1.

Fig. 4 . 1 .

 41 Fig. 4.1. The effect of different tuning methodologies for n = 1800 rpm

Fig. 4 . 2 .

 42 Fig. 4.2. Continuous optimal tuning: a) first pitch angle φp,1, b) corresponding stability charts.

Table 2 . 1 .

 21 Technological parameters of the cutting process under consideration

	Z	κ (deg)	η (deg)	ar (mm)	feed direction
	4	90	0	1		(1, 0, 0)
	Kc,t	Kc,r	Kc,a	ωn,	x	ξx	kx
	(MPa)	(1)	(1)	(Hz)	(%)	(N/μm)
	804	0.314	0.150	178	0.54	19.78

Table 4 . 1 .

 41 The results of different tuning techniques for

	n = 1800 rpm
	tuning technique	pitch angles, φp,i (deg)
	BS tuning	75.4180, 104.5820, 75.4180, 104.5820
	E tuning	87.4852, 92.5148, 87.4852, 92.5148
	BFI tuning	50.5373, 129.4627, 50.5373, 129.4627

Acknowledgement

European Research Council under the European Union's Seventh Framework Programme (FP/ 2007-2013)/ERC Advanced grant agreement no. 340889. Hungarian Scientific Research Foundation OTKA (K108779). EU H2020 MMTECH project (633776 / H2020-MG-2014).