Dynamic modeling and stability analysis for the combined milling system with variable pitch cutter and spindle speed variation

Accepted Manuscript

Dynamic modelling and stability analysis for the combined milling system with variable pitch cutter and spindle speed variation Gang Jin , Houjun Qi , Zhanjie Li , Jianxin Han PII: DOI: Reference:

S1007-5704(18)30087-X 10.1016/j.cnsns.2018.03.004 CNSNS 4476

To appear in:

Communications in Nonlinear Science and Numerical Simulation

Received date: Revised date: Accepted date:

18 August 2017 17 November 2017 3 March 2018

Please cite this article as: Gang Jin , Houjun Qi , Zhanjie Li , Jianxin Han , Dynamic modelling and stability analysis for the combined milling system with variable pitch cutter and spindle speed variation, Communications in Nonlinear Science and Numerical Simulation (2018), doi: 10.1016/j.cnsns.2018.03.004

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ACCEPTED MANUSCRIPT Highlights 

A milling dynamic model, which can simultaneously take into account the effect of the variations of cutter pitch angles and spindle speed, is constructed.

 Through comparisons with previously associated works along with time-domain simulations, the effectiveness of the proposed model is confirmed. The combined influences of the new system and the main system parameters on stability are deeply evaluated and discussed by conducting lots of simulations. Results show that the combined milling process exhibits a great capability to avoid the onset of milling chatter and can result in nearly 3-fold increase in depth of cut than that of a tradition one, but 1-fold increase for the process with variable pitch cutter or variable spindle speed in some special

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cases.

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

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Dynamic modelling and stability analysis for the combined milling system with variable pitch cutter and spindle speed variation Gang Jina,b, Houjun Qia,b , Zhanjie Lia,b, Jianxin Hana,b, a

Tianjin Key Laboratory of High Speed Cutting and Precision Machining, Tianjin University of

b

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Technology and Education, Tianjin, 300222, China National-Local Joint Engineering Laboratory of intelligent manufacturing oriented automobile die

& mould, Tianjin University of Technology and Education, Tianjin, 300222, China

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Abstract

Variable-pitch cutter and spindle speed variation are two well-known methods to suppress milling chatter by disrupting regenerative effect. Considering their compatibility in practical application, their combined system perhaps has better potential for chatter suppression. However, to authors‟

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knowledge, there has been little research associated with this issue. In this paper, a milling dynamic model, which can simultaneously take into account the effect of the variations of cutter pitch angles

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and spindle speed, is constructed and linear stability analyses are carried out via an updated

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semi-discretization method, which has great capability to predict the stability lobes of milling process with multiple time-periodic delays. Then, through comparisons with previously associated

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works along with time-domain simulations, the effectiveness of the proposed model is confirmed. Moreover, the combined influences of the new system and the main system parameters on stability

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are deeply evaluated and discussed by conducting lots of simulations. Results show that the combined milling process exhibits a great capability to avoid the onset of milling chatter and can result in nearly 3-fold increase in depth of cut than that of a tradition one, but 1-fold increase for the process with variable pitch cutter or variable spindle speed in some special cases. Key words: Milling chatter; Variable pitch cutter; Spindle speed variation; Combined effect 

Tel.: +86 2288181083 Email address: [email protected]

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ACCEPTED MANUSCRIPT 1. Introduction Machining by material removal is one of the most widely used manufacturing processes. However, due to the instability of a cutting process known as chatter, high material removal rates often cannot be achieved in practice. Besides, large dynamic loads caused by chatter vibrations also damage the machine spindle, cutting tool and workpiece. The most powerful source of chatter is regeneration. It

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is associated with the dynamics of the machine tool and the cutting force variation due to subsequent cuts on the same cutting surface. Starting with regeneration mechanism, lots of researchers paid great effort for enhancing milling stability in the past few decades.

The established method for predicting and avoiding regenerative chatter is the one that chooses

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cutting parameters from the stability lobe diagram (SLD). Based on the obtained SLD, the stable milling operations can be distinguished from the unstable ones and then chatter-free milling operations are determined. So far, various typical methods, including the analytical, numerical and experimental ones have been proposed to predict SLD, e.g., the zeroth-order approximation method

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[1], multi-frequency solution [2], temporal finite element analysis [3], Chebyshev polynomial

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approximation [4,5], semi-discretization method (SDM) [6-8], full-discretization method [9,10], spectral element method [10, 11], Galerkin Approximations [13] and so on.

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As known, the above methods aim at reducing or avoiding chatter by choosing chatter-free

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parameters from SLD. However, some methods focused on enlarging the stable zone of the SLD by disrupting the regenerative effect. Typical example is the utilization of variable pitch cutter (VPC).

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Compared with constant pitch cutter (CPC), its phase between two waves is not constant for all teeth, thus the modulation in chip thickness is achieved and vibration under disturbance will be reduced. The idea of VPC was first suggested by Slavicek [14]. Altintas et al. [15] and Budak [16, 17] adapted an analytical milling stability model for variable pitch cutters in frequency domain. In order to consider the cutter runout or variable pitch angle, a unified SDM [18], an improved FDM [19, 20] have been proposed successively for predicting the stability lobes of milling process with multiple delays. Dombovari et al. [21] investigated the mechanics, dynamics and stability models of milling 3

ACCEPTED MANUSCRIPT with serrated cutters. Results show that serrated cutters can result in higher stable depth of cuts compared with their non-serrated counterparts. For variable-pitch or variable-helix milling, Sims et al. [22], Dombovari and Stepan [23], and Jin et al. [24, 25] used improved SDM to examine the effect of the tool geometries on the stability trends. Recently, Stepan et al. [26] compared the stability behaviour of milling processes performed by conventional, variable helix and serrated milling tools,

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and pointed out that for roughing operations, the highest stability gain can be achieved by serrated cutters, but variable helix milling tools can achieve better stability behaviour only if their geometry is optimized for the given cutting operation. Jin et al. [27], Sims [28] and Guo et al. [29] proposed efficient methods to predict the stability for variable helix milling.

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Variable spindle speed (VSS) is another alternative method for regenerative chatter disruption. It has similar mechanism to the use of cutter with variable pitch or variable helix. However, its positive effect is due to the reason that the phase relationship between the inner and outer modulation on the machined surface is changed by modulating traditional constant spindle speed (CSS) periodically.

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The limitations of this method are the responsiveness of machine tools and the load on spindle motors

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not designed to constantly change spindle speed [30]. The idea of suppressing chatter by spindle speed variation came up first in the1970s. Takemura et al.

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[31] analyzed the stability of turning with variable speed by means of an energy balance for triangular, rectangular and sinusoidal speed variations. They found the amplitude of chatter vibrations can be

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reduced by using VSS. Inamura and Sata [32], Sexton and Stone [33], and Tsao et al. [34] all

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conducted the experiments and showed that the chatter could be greatly suppressed when suitable amplitude and frequency are chosen. Insperger and Stepan [35] used the SDM to construct stability diagrams for variable speed turning and revealed the increase of the critical depths of cut in the low speeds. Zatarain et al. [36] presented a general method in the frequency domain, and pointed out that varying spindle speed can effectively be used for chatter suppression with low cutting speeds. Seguy et al. [37] studied VSS in high speed milling and found that the stability properties can always be improved by spindle speed variation within the unstable domain of the first flip lobe. Recently, Otto 4

ACCEPTED MANUSCRIPT and Radons [38] investigated the properties of the time varying delay dependent on a modulation of the spindle speed. They pointed out that a large stabilization of the process is possible via the implementation of a suitable VSS. Xie and Zhang [39], Totis et al. [40] and Niu et al. [41] efficiently evaluated the process stability in milling with spindle speed variation via improved SDM, Chebyshev collocation method and variable-step numerical integration method, respectively. Through simulation

spindle speed variation technique in milling process.

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analysis and experimental tests, Bediaga et al. [42] investigated the effectiveness of continuous

Table 1 Different milling processes Spindle Speed

Cutter

VPC

CSS

Process I

Process II

VSS

Process III

Process IV

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CPC

It can be clearly seen from Fig. 1 that the spindle speed as well as the cutter pitch angles could be

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constant or variable for a milling system. Starting with this consideration, four different milling processes, i.e., „CPC and CSS‟, „VPC and CSS‟, „CPC and VSS‟ and „VPC and VSS‟ can be

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constructed. Conveniently, they are respectively named by Process I, II, III and IV, as shown in

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Table 1. Looking back to the references introduced before, it can be found that Ref. [1-13, 29] focused on the milling stability for Process I, while Ref. [14-28] paid great attention to the milling

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with Process II, but the milling stability associated with Process III was the research emphasis and topic for references [31-42].

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For Process IV, however, there has been little involved research to authors‟ knowledge. From a

practical view, it is easily known that one machine tool spindle can clamp a cutter with variable pitch angles while its speed is modulated periodically in milling process. Therefore, one can say that Process IV is an exercisable operation. On the other hand, two interesting and thought-provoking questions arise and should be deeply discussed and solved: (1) Does Process IV make much more sense for gaining a better capability of chatter suppression, compared to Processes I, II and III? 5

ACCEPTED MANUSCRIPT (2) If Process IV really works, it is needed to know how it influences on the milling stability and how to achieve a better stability gain. With above questions in mind, in this paper, one process dynamics model, which can simultaneously take the features of variable-pitch cutter and spindle speed variation into account is built, and then some analyses are conducted. The focus of the current work is to exploit the combined

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potential of VPC and VSS for chatter suppression. The structure of the paper is as follows. In Section 2, the mathematical model is introduced. In Section 3, the numerical algorithms are described in detail and the stability analysis is conducted along with method verification. In Section 4, some

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discussions are given. In Section 5, conclusions are presented in brief.

milling process.

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Fig 1. Schematic drawing of the combination of spindle speed variation and variable pitch cutter in

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2 Modelling for Process IV

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In this section, the parameters relevant to spindle speed variation and variable pitch cutter are defined, and then the mechanical model for Process IV is presented. 2.1 Form of VPC and VSS 2.1.1 VPC In [16, 17], two types of VPCs are reported. One of them is a linear tooth pitch variation, which can lead to a higher stability gain, compared to a regular pitched end mill. The linear tooth pitch variation 6

ACCEPTED MANUSCRIPT has the combination of pitch angles as:

  [1, 2 , , N 1, N ]  [ 0 , 0   , , 0  ( N 1) ]

(1)

where  j is the pitch angle between the subsequent j-1th and the jth flutes in degree (cf. Fig. 1b), N is the number of teeth. For one VPC, whose pitches are varied in step of a pitch increment  , the

0 

360 ( N  1)  N 2

2.1.2 VSS

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basic pitch variation  0 can be obtained by (2)

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The spindle speed can keep constant or be modulated periodically around a mean value by a given amplitude and frequency in milling process, such as sinusoidal, triangular or square-wave modulations [37]. In this paper, one of periodic spindle speed variation, the sinusoidal modulation will be considered and analyzed. As shown in Fig. 2, a sinusoidal spindle speed variation (t ) is

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periodic at a time period T  60 / 0 / RVF , with a nominal value 0 , and an amplitude

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1  RVA 0 . For this sinusoidal modulation, the shape function is modeled as: 2 2 t + )  0 [1  RVA  sin(RVF  0t + )] T 60

(3)

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(t )  0  1 sin(

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where RVA  1 0 is the ratio of the speed variation amplitude to the nominal spindle speed, and

RVF  60 (0T ) is the ratio of the speed variation frequency to the nominal spindle speed.  is the

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phase angle of the sinusoidal modulation at time t =0. Here, it should be noted that to simplify the analysis in the following, it is assumed that as the 1st

tooth of the cutter starts to rotate from the time t =0, the spindle speed is modulated from the phase angle  =0.

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Fig 2. Schematic drawing of the sinusoidal modulation of the spindle speed

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2.2 Structural model

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Fig 3. Schematic drawing of the sinusoidal modulation of the spindle speed

In the chip removal process for a milling operation, the relatively more flexible part (tool or

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workpiece) will experience bending vibrations and leaves a wavy surface behind. This system can be modeled as an oscillator excited by the cutting force. One of the simplest models [6] is shown in the

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left graphs of Fig. 3. The workpiece is assumed to have flexibility in y directions with modal mass m , damping coefficient c and spring stiffness k , while the tool is assumed to be rigid. The straight fluted cutter with radius R and N unequally spaced teeth (indexed by j=1, 2,…, N) is rotating at a variable velocity (t ) in rpm. According to Newton‟s law, the motion equation of the workpiece reads:

my(t )  cy(t )  ky(t )  Fy (t )

(4) 8

ACCEPTED MANUSCRIPT where y (t ) and Fy (t ) are the position and cutting force in the y direction. 2.2.1 Calculation of cutting force Fy (t ) Milling forces can be modeled for given cutter geometry, cutting conditions, and work material. Consider the cross-sectional view of Process IV shown in Fig. 3. f j (t )   f j ,t (t )

T

f j ,r (t )  is the

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cutting force vector for the j-th flute in tangential (t) and radial (r) directions. For a mechanistic force model [16], it can be expressed as a linear or nonlinear function of the axial depth of cut a p and the chip thickness h j (t ) [1]. Here, the dependence of f j (t ) on a p and h j (t ) is assumed to be nonlinear, and consequently it can be expressed as:

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f j (t )  g ( j (t ))a p hqj (t ) K

(5)

with K  [kt , kr ]T is the cutting force coefficients matrices, that can be obtained experimentally [6],

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q is the cutting-force exponent, g j (t ) denotes whether the j-th tooth is cutting, which is given by:

(6)

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1 if st  ( j (t ) mod 2 )  ex g j (t )   0 otherwise

where  j (t ) is the angular position of the j-th tooth. Generally,  j (t ) is the function of helix angle

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[18], thus, the helix angle is always used to determine  j (t ) to exhibit its effect on milling process

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stability. For a helical tool, the forces acting on cutting edges change along the axial direction of the

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tool, and  j (t ) will change with helix angle. In the case of large axial immersion, this effect can be important. Ref. [43] pointed out that the helix has an important influence on flip lobes, while the influence on the traditional lobes can be negligible. However, because the straight fluted mill (with 0° helix angles) is chosen here, it can be defined by:

 2 t ( s)ds   60 0  j (t )   j -1 t  2 ( s)ds   2 i   i 1 360  60 0

if j  1 (7)

if 1  j  N

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ACCEPTED MANUSCRIPT Considering the sinusoidal modulation of spindle speed as described in Eq. (3) and  =0,  j (t ) in Eq. (7) can be expressed as:

60 RVA 2  2 ( 0 t  [1  cos( t )])  2 RVF T  60  j (t )   j -1  2 (0t  60 RVA [1  cos( 2 t )])   2 i  60 2 RVF T i 1 360 

if j  1 (8)

if 1  j  N

in cut yield N

Fy (t )   T j (t ) f j (t ) j 1

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Then, projecting the cutting force f j (t ) in y directions and summing the contributions of all flutes

(9)

form   sin  j (t ) cos  j (t )  .

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2.2.2 Calculation of chip thickness h j (t )

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where Tj (t ) is the transformation matrix between the (t-r) and (x-y) coordinate systems, and has the

When cutting force creates a relative displacement between tool and work-piece at a cutting point,

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the chip thickness experiences the waves on its inner and outer surfaces caused by the present and

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past vibrations. As shown in the right graphs of Fig. 3, the value of the chip thickness h j (t ) is

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composed of static chip thickness h stj (t ) , the dynamic displacements v j (t ) and v j 1 (t ) corresponding to the present tooth j and previous tooth j-1 in the radial direction. Thus, h j (t ) can be

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expressed as

h j (t )  h stj (t )  (v j 1 (t )  v j (t ))

(10)

hstj (t )  f j ,VV (t ) sin  j (t ), v j 1 (t )   y(t   j (t )) cos  j (t ), v j (t )   y(t ) cos  j (t )

(11)

with

where  j (t ) represents the time delay between the teeth j and j-1 due to spindle rotation and variable pitch angles. Its detailed analysis will be described in Section 2.2.3. f j ,VV (t ) represents the feed per 10

ACCEPTED MANUSCRIPT tooth under the circumstance of Process IV, which can be described by f j ,VV (t )  v f  j (t ) 

fz N  j (t ) 60

(12)

where v f is the feed velocity and equals f z N 60 , and f z is the mean feed per tooth. 2.2.3 Calculation of time delays  j (t )

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Time delay is one of the most important intrinsic components of machine tool chatter modeling [44]. Taking into consideration of the non-ideal feature and the utilization of unconventional tools or spindle speed for a milling system, the associated time delays maybe various in form, such as the case of multiple time-delays associated with variable pitch cutter [15-17, 20, 24-26] or cutter runout

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[18], the distributed case associated with variable helix tools [21, 23], the time-dependent case corresponding to spindle speed variation [35, 37, 39] or feed rate effect [45], the state-dependent case corresponding to trochoidal path of the teeth [46].

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Here, an ideal milling process is considered. That is, it is not affected by the runout and trochoidal path mentioned above. In this case, the time delay can be determined by calculating the

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rotation time of the tool over the pitch between two successive teeth [16]. Fig. 4 shows the cutting trajectories generated by the different teeth of a four-fluted cutter. As the j-1th tooth rotates a pitch

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angle  j at (t ) from point cj-1 and the jth tooth reaches point cj, the tool center moves from

For a milling system with equally pitched tool rotating at constant  (i.e., Process I), the time

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

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points oj to oj-1 in a time  j . As a result,

delays are equal and constant (see Fig. 4b), and can be expressed as





60 N

(13)

For a milling system with unequally pitched tool rotating at constant  (i.e., Process II), the

time delays are dependent on the value of  j and the system experiences multiple time delays (see Fig. 4c). Note the maximum number of time delays is N and its actual value is decided by the 11

ACCEPTED MANUSCRIPT number of the different  j . In this case, the related time delays can be expressed as

j 

j 360

0 

 j 60 360 



j

(14)

6

where  0 is the period of each tool revolution. 

For a milling system with unequally pitched tool rotating at inconstant (t ) (i.e., Process IV),

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besides the different values of  j , (t ) also affects the final time to pass  j . Therefore, every delay corresponding to  j becomes a function of t (see Fig. 4d). Based on the idea in Ref. [35] and considered the Eq. (3), the time delays in this case can be given in the implicit form:

t

j ( s) ds  j (t ) 60 360

(15)

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t



Substituting Eq. (4) into Eq. (15), the following relationship can be obtained: 1

m

cos(m (t   j (t ))) 

1

m

cos(mt ) 

j 6

(16)

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0 j (t ) 

Obviously,  j (t ) in Eq. (16) cannot be given in closed form and can only be computed

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numerically. However, if 1 is small enough compared with 0 , i.e., RVA is small, Eq. (16) can

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be expanded and the approximated form of  j (t ) , thus, can be obtained by:

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 j (t )   0, j   1, j sin(

2 2 t )   0, j [1  RVA  sin( RVF  0t )] T 60

(17)

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where  0, j   j / 0 / 6 and 1, j   0, j RVA

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Fig 4. Schematic drawing of construction of time delays in different milling case. (Taking a four-fluted tool as example)

In order to evaluate the effectiveness of the approximation in Eq. (17), two milling processes with

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the same two-fluted VPC ( =[80,100]°) and RVF=0.1 but with different RVA=0.1 and 0.2 under

0 =8000rpm are taken as example. Fig. 5 shows the sinusoidal spindle speed variations as well as

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the correspondingly exact results of the time-delay variations obtained by numerical calculation and

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the approximate one by Eq. (17). It can be seen from Fig. 5 that there exists two time-dependent delays, which are different in amplitude but both periodic at the spindle modulation period T (cf.

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the right graphs of Fig. 5). Clearly, the two delays respectively correspond to the first- and second-tooth, and their ratio is equal to that of the corresponding pitch angles as known from Eq.

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(17). The appearance of the two time-dependent delays is attributed to the combined effect of the pitch angle and the spindle speed variations. On the other hand, as shown in the right graphs of Fig. 5, there is a good consistency between the

exact and approximate values. This indicates the effectiveness of the approximation in Eq. (17). However, it should be noted that compared with the results for RVA=0.1, the case of RVA=0.2 results in larger approximated error in amplitude. This is due to the reason that the approximated condition of Eq. (17) is small RVA. 13

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Fig.5 Sinusoidal spindle speed variations with the corresponding exact and the approximate time-delay variations for 0 =8000 rpm, RVF=0.1 and RVA=0.1 and 0.2.

With the substitution of Eqs. (5), (6), (10) and (11) into Eq. (9), the cutting forces can be

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described as N

Fx (t )   H j (t )( f j ,VV (t )sin  j (t )  cos  j (t )( y (t )  y(t   j (t )))) q j 1

(18)

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with H j (t )  a p g j (t )(kt sin  j (t )  kr cos  j (t ))

Substituting Eqs. (17) and (18) into Eq. (4), and linearizing the consequent equation on the basis

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of the assumption that the motion of the workpiece can be written in the form (19)

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y(t )  y p (t )  q(t )

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where x p (t ) is the nonhomogeneous solution and q(t ) is the infinitesimal perturbation [6], then a linear time-periodic DDE can be given by N

(20)

1 G j (t )  g j (t )a p qf jq,VV (t )(kt sin q  j (t ) cos  j (t )  kr sin q 1  j (t ) cos 2  j (t ))

(21)

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mq(t )  cq(t )  kq(t )   G j (t )(q(t )  q(t   j (t ))) j 1

with

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ACCEPTED MANUSCRIPT 3 Stability of milling with VSS and VPC 3.1 Stability analysis with an updated SDM Obviously, the governing equation shown in Eq. (20) is a typical second–order DDE with multiple and time-periodic delays. By introducing the system vector u(t )  (q(t ), q(t ))T and using Cauchy transformation, the order of Eq. (20) can be halved, and written in the first-order form as N

j 1

0  0 N    , B(t )  C (t ) G ( t ) C ( t )  ,  j j  j 0 2 xnx  j 1  mx 

1

(22)

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 0 with A   2 nx

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u(t )  Au(t )  B(t )u(t )   C j (t )u(t   j (t ))

where n  k m and   c (2n m ) are is the angular natural frequency and the relative damping.

For a periodic DDE, the Floquet theory can be applied to determine stability properties. To

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ensure that Eq. (22) is periodic, the spindle modulation period T  60 / 0 / RVF and the rotation

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period  0  60 / 0 corresponding to VPC must satisfy one basic assumption [6]: the ratio of

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 0 / T is a rational number, i.e., q1T  q2 0 with q1 and q2 are relative prime numbers. In this case, q1 / q2 equals the value of RVF, and the principal period of this system is q1T . In the

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following, an updated SDM based on the Floquet theory is proposed to solve equation (22) to

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determine the stability lobes. The first step of semi-discretization is the construction of the discrete time interval [ti , ti 1 ] with

length t , i  0,1, , i.e., the system period T (defined q1 =1) is divided into k number of discrete time intervals, and resulting in T  k t . Then, t can be calculated by

t 

T 60  k k  RVF  0

(23)

Considering Eq. (17), the average delay for the discretization interval [ti , ti 1 ] is defined as 15

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 i, j 

1 ti1  j (t )dt   0, j   1, j ci t ti

(24)

where ci 

k 2



( i 1)2 / k

sin(t )dt , i  0,1,

i 2 / k

, k  1.

(25)

 i , j  t / 2

mi , j  int(

t

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The number of intervals mi , j related to the delay item  i , j can be approximately obtained by

 j  k  RVF  (1  RVA  ci )

)  int(

360

 0.5)

(26)

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where int (*) indicates the operation that rounds positive number towards zero. Then, the maximum value of mi , j is defined by

M  max {mi , j } i , j 1,2,

(27)

M

Considering Eq. (24), Eq. (22) in interval [ti , ti 1 ] can be approximated as [9] N

u(t )  Au(t )  B(t )u(t )   C j (t )u(t   i , j )

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where

(28)

j 1

Bi 1  Bi (t  ti ), t Ci 1, j  Ci , j C j (t )  Ci , j  (t  ti ), t u u u(t )  ui  i 1 i (t  ti ), t u(t   i , j )  i , j ui mi , j   i , j ui 1mi , j

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B (t )  Bi 

(29)

where Bi  B(ti ) , Ci , j  C j (ti ) , ui  u(ti ) , i , j  (mi , j t  t / 2   i , j ) t and i , j  1  i , j . Solving Eq. (28) as an ordinary differential equation over the discretization period [ti , ti 1 ] , one end up with N

ui 1  (0  Fi )ui  Pu i i 1   ( i , j Ri , j ui  mi , j   i , j Ri , j ui 1 mi , j ),

(30)

j 1

16

ACCEPTED MANUSCRIPT where 2 1 1 1 2  2 3 ) Bi  ( 2  2 3 ) Bi 1 , t t t t 1 1 1 Pi  ( 2  2 3 ) Bi  ( 2 3 ) Bi 1 , t t t 1 1 Ri , j  (1  2 )Ci , j  ( 2 )Ci 1, j . t t

Fi  (1 

(31)

0  e At , t

1   e A ( t  s ) ds  A1 (0  I), 0

t

2   se A ( t  s ) ds  A1 (1  tI), 0

t

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Clearly, 0 , 1 , 2 and 3 can be expressed as follows

(32)

3   s 2 e A ( t  s ) ds  A1 (2 2  t 2 I),

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0

where I denotes the identity matrix. State augmentation of Eq. (30) results in the 2(M+1) dimensional discrete map:

Vi +1  ZiVi

M

(33)

with

0 0

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I

0

0 0 0 0  0 0 0 0  0 0 0  N 0     j =1  0 I 0 0   0 I 0 0

 H i 1 i , j Ri , j

 H i 1i , j Ri , j

0

0

0

0

0

0

0

0

0 0  0  (35)  0  0

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0

(34)

, ui M +1 ,ui M )

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 H i 1 ( 0  Fi )  I   0 Zi =    0  0 

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Vi  col (ui , ui 1 ,

where Hi 1  ( I  Pi )1 . The horizontal position of the discrete input matrices Hi 1i , j Ri , j and

Hi 1i , j Ri , j in Eq. (35) depends on the value of mi , j corresponding to  i , j and respectively begin from the column of 2 mi , j -1 and 2 mi , j +1. The approximate Floquet transition matrix is obtained as

Φ  Zk 1 Zk 2

Z1Z0 . If the moduli of all the eigenvalues of the transition matrix Φ are less than

unity, the system is stable. Otherwise, it is unstable. 17

ACCEPTED MANUSCRIPT Here, it also should be noted that as it was shown in Refs. [6, 8], one reduction about the matrix

Vi can be carried out because only the delayed positions show up in the governing equation of the milling process. Thus, the size of the approximation vector in Eq. (34) could be reduced by removing the delayed values of the velocities, such that the size of vector Vi can be decreased to

M  2 , i.e., Vi  col(qi , qi , qi 1 , qi 2 ,

,qi M ) . This can give some additional improvement in the

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computational time for the proposed method. 3.2 Analysis of stability influence

In this section, two key problems involved in the Process IV influences on milling stability and

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the VSS and VPC parameters influence on the stability of Process IV will be investigated and answered. Around this topic, groups of simulation examples are designed, and the associated stability charts are calculated by the proposed method. In the following simulations, excepted for the calculated parameters associated with VPC and VSS, the other parameters are almost the same.

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They are: up-milling, half-immersion, the number of the cutter teeth is N =4, the cutting-force

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coefficients are Kt  107 106 N/m1q and Kr  40 106 N/m1q , where q is the cutting-force exponent and is equal to 0.75, the mean feed per tooth is f z =0.1mm. The mode mass is

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m  3.1663 kg, the natural frequency is n =400 Hz, damping ratios is   0.02 . Note that m ,

speed.

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n and  are assumed keeping constant with the change of the pitch angles and the spindle

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Besides, to quantitatively evaluate the stability gain caused by certain parameter, two variations

called stable area ratio  and stability gain rate  are defined. The former represents the ratio between the values of the stable area A0 and the total area A for a selected parameter plane of the spindle speed and the axial depth of cut, and can be expressed as

=

A0 A

(36)

Obviously, the bigger value of  means the more stable area. The latter shows the comparison 18

ACCEPTED MANUSCRIPT of the stable area ratios corresponding to two cases. For example, compared case “B” with “A”, it can be represented as

 =

 B - A 100 A

(37)

3.2.1 Influence of process IV on stability

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Figs. 6 and 7 illustrate a number of stability charts in the high- and low-speed domain, respectively. The stability boundaries represented by black solid lines correspond to Process I for reference. The stability lobes corresponding to Process II with  =[75, 85, 95, 105]° (  =10°) are shown by blue solid lines while the stability charts for Process III under the value of RVA=0.1

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but different values of RVF are plotted by red dashed lines. The green thick lines show the stability boundaries related to Process IV, which are constructed on the basis of the parameter combination of above Processes II and III. It can be seen clearly from Figs. 6 and 7 that compared with Process

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I, Processes II and III can result in some improvements in the stability. Clearly, these improvements are more significant for low-speed domain. Above phenomenon shows the active

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influence of variable pitch cutter and spindle speed modulation on milling stability through regenerative chatter disruption, and the results are consistent with relevant prior works [6, 16, 22-24,

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35]. Then, a VPC is utilized in Process III or spindle speed modulation is applied for Process II, the

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original system will be converted into a new one with Process IV. From every subgraph of Figs. 6 and 7, one can see that clear deviations exist for their stability boundaries. Compared with Process

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III, Process IV nearly results in always higher axial depth of cut for the selected spindle speed domain. As to Process II, Process IV still can obtain higher axial depth of cut for the selected speed, except for very few regions like  =900 or 1200rpm in low-speed domain or  =6000rpm in high-speed domain. However, comparing the high-speed domains, the influence of Process IV on milling stability seems more significant in low speeds. For instance, some corresponding local minima of axial depth of cut, e.g., the values for  =13000 in Figs. 7(b)-(d), can reach up to 10.1, 10.2 and 9.9 mm, which are nearly 4-fold as high as that for process I and are about 2-fold for 19

ACCEPTED MANUSCRIPT processes II and III at this moment. Thus, the combined effect of Process IV on improving the stability region and suppressing milling chatter is exhibited and illustrated. To exhibit the influence of above three processes on stability quantitatively, the stable area ratios

 (shown in Eq. 36) corresponding to the stability charts in Figs. 6 and 7 are calculated and shown in Figs. 8 and 9, respectively. As can be seen, the obtained values of  , obviously, keep increasing

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for Processes from I to III to II to IV. Moreover, compared with Processes I, II and III, Process IV could gain the remarkably bigger  , especially for low-speed domain. As shown in the right graphic of Fig. 9, the biggest value of  occurs in the case of Process IV with RVF=0.05 and equals 0.533, if taking process III as references (about  =0.238), its stability gain (see Eq. 37) is

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about  =128％, which indicates the utilization of a VPC can increase of the stable area by 128％ on the basis of process III. To sum, Process IV has more significant effect on stability gain than processes I, II and III, and it is a better method to obtain possibly large stable area.

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Here, it should be noted that the stability boundaries colored as black and red in Figs. 6 and 7 are in close agreement with the results obtained by SDM and published in [6] (for details see Figs 5.17

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and 5.18 in Chapter 8) under the same parameters, this indicates the effectiveness of our method in

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predicting milling stability for Processes I and III.

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Fig. 6 Comparison of the stability chart corresponding to Process I, II, III and IV in the high-speed domain, where Processes III and IV are constructed under different RVF (Note the values of RVF for process I and II equal 0): (a) RVF=0.5; (b) RVF=0.2; (c) RVF=0.1 and (d) RVF=0.05. For interpretation of the references to colour in this figure legend, the reader is referred to the web

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version of this article.

Fig. 7 Comparison of the stability chart corresponding to Process I, II, III and IV in the low-speed domain, where Processes III and IV are constructed under different RVF (Note the values of RVF for process I and II equal 0): (a) RVF=0.5; (b) RVF=0.2; (c) RVF=0.1 and (d) RVF=0.05. 21

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Fig. 8 Comparison of the values of  corresponding to the stability charts in Fig. 6.

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Fig. 9 Comparison of the values of  corresponding to the stability charts in Fig. 7.

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3.2.2 Influence of VSS parameters on the process IV The VSS parameters, RVF and RVA associated with the sinusoidal modulation of spindle speed

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as shown in section 2.1.2 are also considered and investigated here.

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Fig. 10 shows the stability charts corresponding to four different RVF under 50% and 10% radial immersions, respectively. It can be seen that as the values of the RVF change from 0.05 to 0.20, nearly no difference among the stability boundaries has occurred. However, as its value is achieved to 0.50, i.e., relatively fast spindle speed modulation, small differences among associated stability charts occur. Unfortunately, the bigger RVF seems result in the relatively lower depth of cut. In short, the influence of RVF on the process IV is small for either high-speed domain or low-speed one. Although a bigger value of RVF somewhat makes it clearer, this is disadvantage for 22

ACCEPTED MANUSCRIPT suppressing chatter. The stability lobes for 50% and 10% radial immersion up milling are constructed in Fig. 11. For every immersion, four operations associated with Process IV are performed by the VPCs with

 =[75, 85, 95, 105]° and RVF=0.20 but four different RVA, i.e., 0.05, 0.10, 0.20 and 0.50. It can be seen from Fig. 11 that the change of RVA can result in a large stability difference. As the

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increasing of RVA, the most stability limits gain obvious improvement to some extent. From RVA=0.05 to 0.50, their corresponding values of  are equal to 0.334, 0.356, 0.387 and 0.539 for 50% immersion (0.643, 0.644, 0.687 and 0.822 for 10% immersion), also keep a clearly increasing trend. As to the reason why larger RVA can result in better stability gain, one can attribute this as

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following. As the RVF is relatively small, the values of  j (t ) in Eq. (17) as well as that of

f j ,VV (t ) in Eq. (12) are also small. In this case, the effect of pitch angles variation on the instantaneous uncut chip thickness h j (t ) will be relatively large. As a result, a stronger effect on

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stability lobes occurs. Besides, it should be noted that as the increase of stability gain, the bigger

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RVA would result in the more flattened maximum stability limits (e.g., the limits around  =3000 and 6000rpm). This phenomenon is the same as that for VSS milling system described in [39].

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Overall, adjusting the RVA and RVF to obtain maximum stability gain is a feasible method for process IV system. However, compared to RVF, RVA has smaller or nearly negligible influence on

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stability gain.

Fig. 10 Comparison of the stability charts corresponding to four different values of RVF for the 23

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process IV. (a) 50% radial immersion; (b) 10% radial immersion.

Fig. 11 Comparison of the stability charts corresponding to four different values of RVA for the

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process IV. (a) 50% radial immersion; (b) 10% radial immersion.

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Fig. 12 Comparison of the stability charts corresponding to three different VPCs for the process IV.

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(a) 50% radial immersion; (b) 10% radial immersion. 3.2.3 Influence of VPC parameters on Process IV

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To illustrate the performance of the variable-pitch feature on dynamic behavior of Process IV,

three different 4-fluted VPCs, respectively with  =5°, 10° and 15° are considered. Their pitch variations are  =[82.5, 87.5, 92.5, 97.5]°, [75, 85, 95, 105]° and [67.5, 82.5, 97.5, 112.5]° on the basis of Eqs. (1) and (2). The VSS parameters are fixed to RVA=0.1 and RVF=0.5. Fig. 12 shows the stability charts for up milling under 50% and 10% radial immersion. As can be seen, significant stability differences occur when the values of  change. Obviously, the

24

ACCEPTED MANUSCRIPT increase of  results in a big improvement for stability limits. Above phenomenon indicates that cutter pitch variations have an advantageous effect on the stability, and the higher  means bigger stability gain, which are consist with the conclusion in Ref. [20] for a VPC milling. However, from a practical perspective,  surely can not be too big because one needs to provide enough room between consecutive teeth for the chip to travel up the flute [47]. Thus, it must be design

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reasonably.

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3.3 Time-domain simulation

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Fig. 13 Stability lobes under four cutting processes and the ten chosen cases.

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Table 2. The cutting conditions corresponding to the ten chosen cases in Fig. 13 Cases A

a p /mm 3.000

 /rpm 1400

Process I

B

3.750

1400

I

C

4.875

1400

I

D

4.875

1400

III

E

5.625

1400

III

F

8.250

1400

III

G

5.625

1400

II

H

8.250

1400

II 25

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8.250

1400

IV

J

8.250

1600

IV

K

9.375

1400

IV

L

10.125

1400

IV

Time-domain simulation is a widely-used and effective approach for milling stability prediction and model validation in the past. From simulation, it is possible to obtain the detailed information

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about the process such as the amplitude of the vibrations, the chip thickness, or the cutting forces [37].

In order to investigate the stability character for milling processes discussed above, one updated time-domain program based on Ref. [48] is crafted. This program can take into account teeth

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jumping out of contact, work-piece dynamics, regeneration effect, pitch angles and spindle speed variations simultaneously. Besides, it is worth noted that in order to compute the position at the previous cut easily, the program takes the spindle speed variation into account by adjusting the time step so that the angular step is kept constant during the simulation [37].

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Fig. 13 illustrates four stability boundaries that respectively corresponds to Processes I, II, III

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and IV (note that these boundaries are same to the ones in Fig. 7a that are framed by a pink rectangle). Meanwhile, considered the stability boundaries by theoretical predictions, ten milling

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cases under different processes and a p around  =1400 and 1600rpm are adopted and labeled by

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solid black points successively in Fig. 13. Emphasizingly, cases A, B and C correspond to Process I, cases D, E and F correspond to Process III, and cases G and H correspond to Process II, while

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cases I, J, K and L correspond to Process IV. For more details about the main parameters related to above cases, please see table 2. Figs. 14 and 15 show the associated details about continuous and 1/ T -sampled vibration

histories, the associated Poincare section and Fast Fourier Transform (FFT) for the cases from A to L by time-domain simulation. Here, it should be pointed out that because the discrete vibration histories corresponding to Processes III and IV are non-uniform in time, the FFT graphs shown in graphics (d) and (f) of Fig. 14 and graphics from (i) to (l) of Fig. 15 cannot be gained on the basis of 26

ACCEPTED MANUSCRIPT the traditional FFT, but nonuniform FFT (NUFFT) [49] because of its great capability of dealing with the discrete non-uniform sample. For a non-uniformly sampled signals y(tn1 ) , NUFFT can be defined by Y (m1 ) 

N n 1

 x(t

n1 0

n1

) exp(

 j 2m1tn1 ) Nn

(38)

coordinates, and m1, n1  0,1,

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where Nn is the total sampling points, m1 and tn1 are the frequency and time sampling

, Nn  1 .

Consequently, the stability states corresponding to above ten cases are as follow:

(1) Case A: one stable process since the 1/tooth passage displacement samples approached a

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single fixed point value (see the middle graphic of Fig. 14a) and there are only tooth passing frequency (TPF) and its harmonics (see the right graphic of Fig. 14a).

(2) Cases B and C: two unstable processes. while the vibration histories for Cases B is divergent

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(see Fig. 14b), the case C are characterized by the appearance of a circle map for Poincare section (see the middle graphic of Fig.14c) that means the quasi-periodic motion associated with a

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secondary Hopf bifurcation [50]. Meanwhile, it can be seen from the right graphic of Figs. 14(b) and 14(c) that the large amplitude vibrations occur at the chatter frequency (CF) of 430.1 Hz.

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(3) Case D: one stable process. It is characteristic by the single fixed point for Poincare section,

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the clearly smaller vibration peaks (compared with Case C) and that only the modulation frequency (MF) and its harmonics appear (see Fig. 14d).

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(4) Cases E and F: two unstable processes, where the quasi-periodic chatter occurs (see Fig. 14e

and Fig. 14f).

(5) Case G and I: two stable processes (see Fig. 15g and I). It should be noted that for case G,

variable pitch angles cause the chip load to be distributed unevenly among the cutting teeth and shifts the frequency content of the cutting force signal away from the TP frequency and towards the spindle rotation frequency (SRF). Thus, only the SRF and its harmonics appear in Fig. 15g. (6) Cases H, J, K and L: four unstable processes, which are represented by the quasi-periodic, 27

ACCEPTED MANUSCRIPT the quasi-periodic, the period-two (characterized by two points in the Poincare section) and the quasi-periodic chatter, respectively (see Figs. 15h, j, k and l). From above simulations, the following results and conclusions can be found: (1) The stability states obtained by theoretical prediction and simulations are in close agreement with each other, even for some cutting conditions around the threshold value by stability prediction,

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like cases B, E, H and K. This indicates the effectiveness and accuracy of our predicted method to a certain degree.

(2) Taking cases B, E, H and K near the stability limits at  =1400 as examples, Process IV can respectively result in almost 200%, 20% and 100% increase in depth of cut than that of Process I, II

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and III. On the other hand, when possess IV is utilized under the same conditions, the unstable cases F and H both become stable process (i.e., case I). Obviously, the better stability gain for

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Process IV has been captured and proved via time-domain simulations.

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Fig. 14 The information associated with the continuous and 1/T-sampled time histories of vibration displacement y, the Poincare section and the Fast Fourier Transform for the cases A-F of Fig. 13.

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(For cases A, B and C, T is the tooth passing period and equal to about 0.0107s as opposed to the modulated period 0.0857s for cases C, D, E and F)

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Fig. 15 The information associated with continuous and 1/T-sampled time histories of workpiece vibration, their Poincare section and Fourier spectrum for the cases G-L of Fig. 13. (For cases from

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I to L, T is the modulated period and equal to 0.0857s as opposed to the spindle rotation period 0.04286s for cases G and H) 4. Discussion

Process IV has exhibited a greatly positive influence on the increase of stability gain and the built theoretical model has shown strong agreement with previously published work as well as with time-domain simulations. However, various aspects of the results are worthy of further discussion. 1. From a stability influence aspect. The combined effect of VPC and VSS on improving the 30

ACCEPTED MANUSCRIPT stability region exists and it is more remarkable in certain cases, such as low spindle speed (see Fig. 7), bigger RVA (see Fig. 11). However, this meaningful effect seems to derive from the both positive influence of VPC and VSS on stability limits directly. In other words, a possible superposition of the helpful influence of VPC and VSS themselves on milling stability has been achieved by utilization of the combing process between them. This may mean that for a combined

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milling system, like Process IV in this paper, its stability properties can be investigated on the basis of the understanding and mastering of that for their involved subsystems.

Obviously, through combining the linear tooth pitch variation with the sinusoidal modulations of spindle speed, in a sense, chatter suppression has been realized through a cooperative control in this

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paper. However, there are still other similar methods which are based on the “variation” or “perturbation” of the regeneration to influence the milling stability. For example, the utilization of the cutters with variable helix angles [22] or serrated flutes [21] and the periodical modulation of spindle speed in the form of triangular or square-wave [37]. So, if some attempts to properly

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combine any two of them in milling process are carried out following the similar consideration in

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this paper, can some schemes offer more superior capability for suppressing chatter? This will be the future work discussed next.

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In addition, VPC and VSS both play important parts in the promotion of milling stability for process IV as shown in Figs. 6 and 7. However, their importance or contribution to the combined

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effect is uncertain. About this question, there is no available method or investigation which can be

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referred to. Nevertheless, from Figs. 8 and 9, it can be seen that compared with process III, the values of  corresponding to process II seem obviously bigger for the chosen cases. This means that process II has more active influence on milling stability than that for process III in some extent. Thus, from this perspective, one can say that for a certain process IV, process II seems play a more important role. In other word, process II could cause most of the improvement of process IV. Of course, above conclusion is obtained only by authors‟ preliminary exploration, deep investigations need to carry out in future. 31

ACCEPTED MANUSCRIPT 2. From a regeneration mechanism aspect. It is well-known that VPC and VSS are effective methods to prevent chatter vibration during milling processes by disrupting the regenerative effect. However, because of their obvious compatibility in practice (see Fig. 1), their combined system has exhibited better capability to suppress milling chatter. In other words, the form of the perturbation of the regeneration has realized upgrading to some extent. This means that as to disrupt regeneration

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in milling process, there may exist some more effective or high-level forms we unknown. Fortunately, one of the possible forms has been constructed here by combing VPC with VSS. However, what is the best or almost best form in them and how to exploit their potentialities to achieve greatest stability gain are the important and difficult issues that is worthy of deep thinking

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and exploration.

3. From a best effect of chatter suppression aspect. Though the combined system of VPC and VSS results in stability gain increasing and shows some regularity to influence the stability region, the adoption of machining parameter for such system to aim at efficient machining operation is still

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a difficult task. For a given milling system, a synchronous optimization of the tool geometry and

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spindle speed to maximize the stability against chatter in machining operation has more practical importance. However, this optimization becomes too complex and tedious since there shows up a

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lot of optimized parameters associated with pitch angle and spindle speed. In this case, optimizing spindle speed or pitch angle but fixing the rest ones may be one feasible policy, and may result in

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better efficiency. Of course, the parameters, which have smooth influence on system stability, such

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as RVF, should be neglected. Finally, although the models have been compared well with previous work on Processes I and III,

and validated against time-domain simulations for Process IV, it is clear that experimental testing is needed. This will be the subject of future work. 5. Conclusion In this work, the process dynamics of milling with variable-pitch cutter and spindle speed variations are investigated. Firstly, a milling dynamic model, which can simultaneously take into 32

ACCEPTED MANUSCRIPT account the effect of the variations of cutter pitch angles and spindle speed, is constructed. Then, linear stability analyses are carried out based on an updated semi-discretization method, and the correctness of the method is verified by the comparisons with previously published works, along with time domain simulations. Moreover, simulations are carried out under certain combinations of parameters to evaluate the combined influence of the new system and the influence of main

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parameters on stability. The following conclusions can be obtained: 1. The combined milling system of VPC and VSS can be modeled by non-autonomous differential equations with multiple and time-periodic delays.

2. The proposed method is effective for predicting stability lobes of milling process with variable

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pitch cutter and variable spindle speed.

3. Process IV exhibits a great capability to avoid the onset of milling chatter. In other words, by combining of VPC and VSS, the form of the perturbation of the regeneration has realized upgrading to some extent.

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4. The depth of cut for Process IV can be nearly 4-fold as high as that for process I and 2-fold for

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processes II and III in some special cutting conditions. 5. The combined effect of VPC and VSS on improving the stability of process IV is more

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remarkable for the cases of lower spindle speed, smaller RVF, and bigger RVA and  , where

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the RVF has a relatively small stability influence. Acknowledge

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This work was supported by the National Natural Science Foundation of China (51405343),

Tianjin Research Program of Application Foundation and Advanced Technology (15JCQNJC05000), Innovation Team Training Plan of Tianjin Universities and colleges (TD12-5043), Tianjin Science and Technology Planning Project (15ZXZNGX00220). Reference [1] Altintas Y, Budak E. Analytical prediction of stability lobes in milling. CIRP Ann-Manuf Techn 1995; 44: 357 – 362. 33

ACCEPTED MANUSCRIPT [2] Merdol SD, Altintas Y. Multi frequency solution of chatter stability for low immersion milling. J Manuf Sci E-T ASME 2004; 126: 459–466. [3] Bayly PV, Halley JE, Mann BP, Davies MA. Stability of interrupted cutting by temporal finite element analysis. J Manuf Sci E-T ASME 2003; 125: 220–225. [4] Butcher EA, Bobrenkov OA. On the Chebyshev spectral continuous time approximation for constant and periodic delay differential equations. Commun Nonlinear Sci 2011; 16: 1541–1554. [5] Butcher EA, Ma H, Bueler E, Averina V, Szabo Z. Stability of time-periodic delaydifferential equations via Chebyshev polynomials. Int J Numer Meth Eng 2004; 59: 895–922.

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[6] Insperger T, Stepan G. Semi-Discretization for Time-Delay Systems Stability and Engineering Applications. 1 ed. Applied Mathematical Sciences. New York: Springer-Verlag. 2011.

[7] Insperger T, Stepan G. Semi-discretization method for delayed systems. Int J Numer Meth Eng 2002; 55: 503–518.

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