Stability Analysis of Milling Process with Variable Spindle Speed and Pitch Angle considering Helix Angle and Process Phase Difference

Suppression of milling chatter by disrupting regenerative eﬀect is a well-known method to obtain higher cutting stability domain. In this paper, a dynamic model of the milling process with variable spindle speed and pitch angle considering helix angle and process phase diﬀerence is presented. Then, an updated semidiscretization method is applied to obtain the stability chart. After the eﬀectiveness of the proposed method is conﬁrmed by comparisons with the previously published works and the time-domain simulations, lots of analyses are conducted to deeply evaluate the inﬂuence of the helix angle, the process phase diﬀerence, and feed per tooth on milling stability. Results show that the change of helix angle can result in signiﬁcant stability discrepancies for both high-speed and low-speed regions. Though the process phase diﬀerence has the randomness and immeasurability in the practical application, it has an important inﬂuence on the stability and will result in a periodic evolution of the stability with a period π . Also, its recommended values are given for the practical milling process.


Introduction
Chatter in metal cutting processes is a form of self-excited vibration and always causes lower material removal rates, machining quality, poor accuracy and surface finish, unpleasant noise and sound, accelerated tool wear, and so on in practice. e most powerful source of chatter is regeneration, which is associated with the surface waviness variation during successive cuts. us, in order to avoid chatter and achieve high productivity in machining, lots of investigations had been done based on the regeneration mechanism for optimizing the machining conditions in the past few decades.
Generally, the regenerative chatter is modeled and analyzed in the frequency domain or time domain. e most established method for predicting and preventing regeneration chatter is of cutting parameters selection from a stability lobe diagram (SLD), which can predict the stability boundary and distinguish the stable and unstable cutting region. Around this theme, various typical methods, such as zeroth-order approximation method [1], temporal finite element analysis [2], multifrequency solution [3], Chebyshev polynomial approximation [4], semidiscretization method (SDM) [5][6][7], full-discretization method (FDM) [8,9], spectral element method [10], have been proposed for more rapidly and accurately predicting the stability chart successively.
Alternatively, some methods focused on enlarging the stable cutting region of the stability chart by disrupting the regenerative effect. One typical example is the utilization of some special cutters, such as variable pitch cutter (VPC) suggested by Slavicek [11]. In this case, the modulation in chip thickness caused by a nonuniform pitch can reduce the vibration under disturbance. Aiming at predicting the stability of milling process with VPC, frequency-domain model [12,13], the Cluster Treatment of Characteristic Roots [14], unified SDM [15], improved FDM [16,17], and improved SDM [18] have been proposed successively. Furthermore, Sims et al. [19], Sims [20], Dombovari and Stepan [21], and Jin et al. [22,23] proposed efficient methods to predict the stability for VPC and variable helix cutter, while Yusoff and Sims [24] optimized the geometry of variable helix tool to suppress regenerative chatter. Besides, the use of serrated cutters [25] also exhibited good capacity and potential for a higher stable depth of cuts.
Another typical example to achieve regenerative chatter disruption is the use of variable spindle speed (VSS), which has a similar mechanism to the use of cutter with variable pitch/helix cutter. Insperger and Stepan [26] and Zatarain et al. [27], respectively, used the SDM and frequency-domain method to construct stability diagrams in the VSS process. Seguy et al. [28] studied VSS in high-speed milling and showed its effective suppression on the unstable domain of the first flip lobe. Long and Balachandran [29] presented a stability treatment for upmilling and downmilling processes with a VSS and discussed the benefits of VSS milling operations. Later, improved SDM [30] and Chebyshev collocation method [31] are presented to efficiently evaluate the process stability in milling with VSS, respectively.
Recently, Jin et al. [32] constructed a new dynamic model to investigate the potential of chatter suppression for the milling process in which the variable pitch cutter and variable spindle speed (VPCVSS) are considered simultaneously. Results show that the combined effect of VPC and VSS on improving the stability region exists and it is more remarkable in certain cases. Obviously, VPCVSS is not a simple combination of VPC and VSS. For modeling, the special characteristics caused by the processes combination, such as time-varying delay, process phase difference, and time-varying feed, significantly increase the difficulty of dynamic modeling, solving, and simulation. For stability impact, the form of the perturbation of the regeneration has realized upgrading to some extent. is upgrade is not a simple superposition of the positive effects of VPC and VSS on system stability but has its own mechanism and phenomenon. However, the research still exists following deficiencies: (1) e actual feature of a tool, helix angle, is not considered during modeling. However, for a general cutting process, the effect of helix angle is crucially important [33,34]. us, the related results cannot reflect the influence of the helix angle on the stability, and its real influence is still worthy of exploration and investigation. (2) e analysis is based on an assumption that the value of the process phase difference is equal to 0; however, it is nonzero usually (note that this will be discussed in detail in Section Section 2.1.1). us, its influence on system stability needs to be explored further. (3) e analysis focuses on the influence of the process parameters, that is, the parameters associated with variable tooth and variable speed, on the system stability. However, other important cutting conditions (such as feed rate, feed direction, and helix angle) are not studied yet.
With the above questions in mind, in this paper, a dynamical model of milling process of VPCVSS considering the process phase difference and the helix angle is established for the first time, and then the influence analyses associated with process phase angle, helix angle, and feed per tooth are conducted based on an updated SDM; some relevant engineering suggestions are given. e structure of the paper is as follows: in Section 2, the mathematical model and the method for milling stability are introduced. In Section 3, the method verification and the stability analysis are conducted. In Section 4, conclusions are presented in brief.

Stability Prediction of Milling with VPCVSS
Generally, the dynamics of the tool-workpiece system in the milling process is described by the models in the form of linear delay-differential equations considering regenerative effect. Based on the number of system modes, these equations usually are single DOF or two DOFs ones. For the former, the workpiece is assumed to be more flexible compared to the rigid tool [5,35,36], but the latter models the tool as a cantilever beam because of the consideration of rigid workpiece [1][2][3].
Here, for the convenience of the following analysis, the single DOF model introduced in [5] is chosen and its schematic diagram is shown in Figure 1(a), where a helix tool with radius R, helix angle β, and N unequally spaced teeth is rotating at a variable velocity Ω(t) in rpm, but the workpiece is excited by the milling force at the tooltip. e mathematical representation of this system is given by where x(t) and F x (t) are the position and cutting force in the x direction. m x , c x , and k x are the modal mass, damping coefficient, and spring stiffness, respectively.

Cutting
Force F x (t). Because of the existence of the cutter helix angle, the height of the cutter teeth in the cutting changes all the time. us, the cutting force acting on the whole axial cutting height is the integration of differential force with respect to the differential axial depth dz along the compliant direction of the workpiece, as shown in Figure 1(c). If the nonlinear force model suggested in [32] is considered, the cutting force here can be expressed as where q is the nonlinear force coefficient. Influenced by the judgment function g(ϕ j ) in equation (2), F x (t) is piecewise continuous [37]. As shown in Figure 1(d), each helix tooth will go through three regions, that is, entry (labeled by ①), middle of the cut (labeled by ②), and exit region (labeled by ③) in a milling process. For one case with the same helix angle and axial depth of cut, there may be two types of such regions due to the different radial depth of cut, as shown in the left and middle graphics of Figure 1(d). Obviously, each region corresponds to different calculation methods of the upper and lower limits of integration in equation (2). Here, function G(ϕ j ) is introduced instead of the original g(ϕ j ), and the unified calculation models for the upper and lower limits of integration; that is, ϕ ej (t) � min(ϕ j (t, 0), ϕ e ) and ϕ sj (t) � max(ϕ j (t, a p ), ϕ s ) are utilized (for more details, see [17,18]). en, equation (2) can be transformed into where c � R/tan β and G(ϕ j (t)) is the new judgment function which determines whether the tooth is in contact with the workpiece, defined by where ϕ s and ϕ e , respectively, are the exit and start angles which depend on radial immersion ratios a D and milling style (up-or downmilling), ϕ hex is the exit angle difference caused by the helix angle and is equal to ϕ hex � a p tan β/R.

Angular Position ϕ j (t).
Generally, angular position ϕ j (t) is the function of helix angle and spindle speed. Here, it is also relevant to the tooth pitch angle. If a linear tooth pitch variation is shown as where ψ 0 � (360/N) − (N − 1)Δψ/2, and a sinusoidal spindle speed modulation modeled as where RVA � Ω 1 /Ω 0 is the ratio of the speed variation amplitude to the nominal spindle speed and RVF � 60/(Ω 0 T m ) is the ratio of the speed variation frequency to the nominal spindle speed, are considered the same as [32], the angular position ϕ j (t) can be defined by As a spindle rotates in the machining process, the pitch angle of VPC changes periodically at the rotation period τ 0 � 60/Ω 0 , whereas the spindle speed varies with the modulation period T m � 60/Ω 0 /RVF. erefore, when VSS and VPC are combined, there must be some problems associated with the phase. As shown in Figure 2, one VPC is rotating with Ω(t), when its 1-th tooth passes through the Waves due to present vibration Waves due to pas tvibration e unfolded current cutting regions positive direction of Y-axis, that is, the red lines; the value of its corresponding instantaneous spindle speed is not equal to Ω 0 , but the value is between intervals [Ω 0 − Ω 1 , Ω 0 + Ω 1 ], which is depending on the phase of Ω(t) at this moment. In order to consider this issue, equation (6) is rewritten in a new form including the initial phase as where η is defined as the process phase difference between VPC and VSS. Obviously, there are kinds of possible η in VPCVSS milling process, such as π/4, π, and 5π/4. In other words, the value of η is a random one between 0 and 2π in the practical application. As a result, the times to pass through any pitch angle ψ j are different when the teeth is rotating at Ω(t, η), for example, Cases (1) and (2) shown in Figure 2. Considering the mechanism of regenerative chatter, it means that η will affect the time-varying characteristics of time delays in the milling system. Further, η does make sense for milling stability.

Instantaneous Chip ickness h j (t).
For a milling process, the instantaneous chip thickness h j (t) generally consists of two parts, that is, the static chip thickness h st j (t) mainly determined by the feed per tooth and angular position, and the dynamic displacements v j (t) and v j− 1 (t), respectively, corresponding to the present tooth j and previous tooth j-1 in the radial direction [1]. us, h j (t) can be expressed as whereτ j (t, η) represents the time delay between the teeth j and j− 1, which is mainly affected by the parameters of VPCVSS, such as spindle speed and pitch angles, and f t,j (t, η) represents the feed per tooth, which can be described by where ϑ j (t, η) � Nψ j (1 − RVA sin(ω m t + η))/360, which is applied to represent the relationship between nominal feed per tooth f m and f t,j (t, η).

Time Delays τ j (t, η).
Generally, time delay is determined by calculating the rotation time of the tool over the pitch between two successive teeth except for some special cases, such as big cutter runout [15,37]. For VPCVSS milling, the time delays can be calculated, based on the idea in [5]. us, an implicit form including τ j (t, η) is conducted as Substituting equation (6) into equation (11) and rearranging both sides of equation (11), then, τ j (t, η) can be approximately expressed by the following relationship: where κ � Ω 0 + Ω 1 sin(ω m t + η). If Ω 1 is small enough compared with Ω 0 , that is, RVA is small, equation (12) can be expanded and then approximated in the form of where τ 0,j � ψ j /Ω 0 /6 and τ 1,j � τ 0,j RVA. Here, it should be noted that the approximation errors of equation (13) increase with the increase of RVA. For small RVA, for example, RVA � 0.2, there are also errors although the effects of errors on the stability lob may be ignored. Substituting equations (7) and (9) into equation (3) and considering equation (1), then linearizing the consequent Start cutting Start cutting Tooth Tooth Figure 2: Schematic diagram of process phase difference for VPCVSS system. equation on the basis of the ideal in [5], a linear timeperiodic DDE can be given by with ε � cq, 2.2. Updated SDM. SDM is a common method often applied to the stability analysis of various DDEs where the time delayed terms are approximated by a piecewise constant function while the current time terms are left unchanged.
Here, the method is presented for the case of VPCVSS. However, to ensure that equation (14) is periodic and applicable to SDM, the ratio of the spindle modulation period T m and the rotation period τ 0 in the current system must be a rational number, that is, q 1 T m � q 2 τ 0 where q 1 and q 2 are relative prime numbers [5]. us, the system period T c can be expressed as By introducing the system vector U(t) � (x(t), _ x(t)) T and using Cauchy transformation, the second-order DDE, that is, equation (14) can be written as a system of the firstorder DDE: with where ω nx � ����� � k x /m x and ζ x � c x /(2ω nx m x ) are the angular natural frequency and the damping ratio. e first step of SDM is the construction of the discrete time interval [t i , t i+1 ] by dividing its period T c into k number with length Δt, that is, Δt � T c /k. en, the time-varying delay τ j (t, η) in equation (17) for the interval [t i , t i+1 ] can be written in an average form based on the ideal in [5] as where en, substituting equation (19) into equation (17), equation (17) can be approximated in [t i , t i+1 ] as new form yields as Δt , where int ( * ) indicates the operation that rounds positive number towards zero. For the initial value U(t i ) � U i , the solution of equation (21) is known: Mathematical Problems in Engineering Substituting t � t i+1 and U(t i+1 ) � U i+1 into the solution leads to Defining the maximum value of m i,j by n � max(m i,j ) and state augmentation of equation (24) results in the 2(n+1) dimensional discrete map: with e horizontal position of the discrete input matrices M i,j,α and M i,j,χ in the right side of equation (27) begins from the column of 2m i,j − 1 and 2m i,j + 1, respectively. Here, it should be noted that if one reduction about the matrix V i is carried out by removing the delayed values of the velocities, like [7], the size of vector V i can be decreased to n + 2, the horizontal position of M i,j,α and M i,j,χ will change from the column of m i,j + 1 and m i,j + 2, respectively. e Floquet transition matrix over the principal period T c is approximated by coupling the solutions of k successive time intervals Δt: Stability of the investigated system is determined by the eigenvalues of the transition matrix Φ. e system is stable if all eigenvalues of Φ are in modulus less than 1.
e main advantage associated with the method in this paper can be summarized as follows: (1) Compared with the original algorithm [5,6,36], the method in this paper can be applied not only to variable pitch milling cutter or variable speed milling process but also to their composite systems. Mathematically, it is suitable for the stability solution of DDEs with multiple variable delays. (2) Due to the addition of equations (16) and (20), the method in this paper is also applicable to some special cases, where number 1 cannot be divisible by the values of RVF, such as RVF � 0.22, 0.3, or 0.4. For the above cases, this method is applicable, but the method in [32] is not.
In addition, it should be noted that the 1-DOF VPCVSS model is proposed in this paper; however, the milling system belongs to the 1-DOF model in general. us, to realize a conversion from 1-DOF one to 2-DOF one [38], some modifications need to be carried out as follows: (a) the system equation (equation (1)) should consider the dynamics in both X and Y directions instead of the original X. (b) Correspondingly, the cutting force model in X and Y directions (equations (2) and (3)) should be constructed. (c) e dynamic cutting force thickness (equation (9)) is changed to a coupled model considering X and Y directions. (d) e system equation (equation (17)) and the corresponding state matrix are changed from two dimensions to four ones. (e) Equations (25) and (26) change from 2(n+1) to 4(n+1) or 2n+4 if the delayed values of the velocities are removed, thus, the horizontal position of the discrete input matrices M i,j,α and M i,j,χ in equation (27), respectively, begins from the column of 4m j,k,i − 3 and 4m j,k,i + 1or 2m j,k,i + 1 and 2m j,k,i + 3.

Stability Analysis and Discuss
In this section, the verification of the proposed method is carried out firstly. en, the influence of the cutting conditions, that is, the process phase angle, the helix angle, and the feed per tooth on the stability of the VPCVSS process is investigated in detail. Note that besides the influence parameters shown in respective figure names, other parameters in the following calculation example are as follows: upmilling, the cutting-force exponent is q � 0.75, the cutting-force coefficients are k t � 107 × 10 6 N/m 1+q and k r � 40 × 10 6 N/m 1+q , the mode mass is m x � 3.1663 kg, the natural frequency is ω n � 400 Hz, damping ratios are ζ x � 0.02, the number of the cutter teeth is N � 4, the tool radius is R � 9.525 × 10 − 3 m, and pitch angles are ψ � [75, 85, 95, 105]°). Here, it must be emphasized and noted that the VPC used in this paper belongs to an asymmetric cutter. Correspondingly, its dynamic balancing ability is poor and this will be more significant in high-speed cutting. However, this problem cannot be considered in the following calculation examples, so there may be some prediction errors associated with cutting stability in the high-speed region.

Model Verification.
In this section, the validity of the proposed method will be carried out by comparing with previous works and the time-domain simulations.
To illustrate the performance of the proposed approach, the related works on VSS in [5] and on VPCVSS in [32] are considered first. Based on the main calculation parameters in [5,32], the proposed method is utilized to calculate the SLDs for both VSS and VPCVSS cases with two different helix angles under two different radial depths of cut. e related results are shown in Figures 3(a)-3(h), where (a)-(d) and (e)-(h), respectively, correspond to VSS and VPCVSS, whilst the stability boundary labeled by blue and red dotted lines are, respectively, associated with the cases of helix angle β � 0°and 30°.
If comparing the SLDs marked by blue lines in  [32], one can see that their calculation results are in good agreement obviously. is indicates that the proposed method has good reliability and accuracy in predicting the stability for both VSS and VPCVSS processes when the helix angle is β � 0°.
Comparing all the blue stability boundaries with the red ones in Figures 3(a)-3(h), it can be found that, no matter for VSS (Figures 3(a)-3

(d)) or VPCVSS (Figures 3(e)-3(h))
milling processes, the change of helix angle can result in significant stability difference.
In order to effectively verify the above predicting trend of stability caused by helix angle, one updated time-domain program (or called time marching solution) is utilized. It should be noted that this program is mainly crafted on the basis of [39], and its application range is expanded from the stability prediction for a single case of VPC to that of VPCVSS. e most critical operation is that the rotation angle is discrete uniformly for one system cycle but time period. Based on the program, detailed information about the amplitude of the vibrations can be obtained and the contour plot of peak-to-peak (PTP) [39] vibration can be generated for a range of spindle speed and axial depth of cut combinations. erefore, the appearance of nonlinear vibration (chatter) can be well grasped by examining the abrupt change of vibration displacement with the axial depth of cut increasing uniformly. Remarkably, this program can take into account teeth jumping out of contact, workpiece dynamics, regeneration effect, pitch angles, helix angles, and spindle speed variations simultaneously. e reason is that the above time-domain simulations can consider the effect of loss-of-contact character on milling dynamics but the proposed method cannot.
In order to detect the stability difference caused by the helix angle more intuitively, two special cutting points are selected, that is, point "A" in Figure 3(g) (Ω 0 � 1600 rpm, a p � 48 mm) and point "B" in Figure 3(h) (Ω 0 � 13000 rpm, a p � 27 mm). One can see from the associated SLDs that points "A" and "B" are unstable in the cases of β � 0°; however, they will turn to the stable ones when the helix angle changes to β � 30°. eir vibration information is simulated in the time domain and the results associated with the cutting vibration time history and periodic points, Poincare section, and nonuniform fast Fourier transform (NuFFT) are shown in Obviously, not to mention CFs, there are so many other frequencies in Figure 5, especially for Figures 5(a) and 5(b). e reason is that the modulated frequency is equal to Ω 0 RVF/(60q 1 ) and its harmonics equal to nΩ 0 RVF/(60q 1 ), with n � 1, 2, . . All in all, the results based on the theoretical calculation and the simulations are consistent.

Influence of the Process Phase
Difference. Because the VPCVSS system is a combination of VPC and VSS, there must be the process phase differences between VPC and VSS as shown in Figure 2. In fact, the value of η is almost impossible to be zero in practice; therefore, whether the conclusion based on the assumption of η � 0 in [32] is universal or not needs further discussion and research.
Starting with above problem, to further evaluate the effect of η on cutting stability, the SLDs corresponding to  four special process phase differences (i.e., η � 0, π/2, π, and 3π/2) under the large and small radial depth of cut, are calculated and shown in Figures 6(a) and 6(b). It can be seen that there are only two stability curves with obvious difference for every SLD: one is the overlapping stability curves corresponding to η � 0 and π (see the blue line and red dotted line) and the other corresponding to η � π/2 and 3π/2 (see the blue line and green dotted line). From a more detailed observation, it can be found that when η changes from 0 to π/2, the system stability is evolving gradually. At η � π/2, the evolution difference reaches the peak. en, η is increasing from π/2 to π, its stability continues to evolve gradually; however, the difference decreases on the whole.
Finally, when η � π, its stability is the same as the case of η � 0. Obviously, there are two kinds of limit shapes of the SLDs, that is, ones at η � 0 or π and the others at η � π/2 or 3π/2. In order to evaluate the impact of η in more detail, four different η are chosen in the interval [0, π/2] corresponding to Figure 6(a) and the interval [π/2, π] corresponding to Figure 6(b), respectively, and the related SLDs are calculated and shown in Figures 6(c) and 6(d). It can be seen that, for the above intervals, η always plays an important role and results in a gradual change of SLDs from one limit to another. If this trend is extended to the whole value domain of η, one can easily know that the limit shapes of the SLDs under the influence of η, respectively, occur at η a � nπ − π and η b � nπ − π/2 (n � 1, 2, ...), and η obviously leads to the periodic evolution of system stability whose period is π. is is attributed to the following reason. For the VPCVSS processes described in this article, their time delays exist in the form of sine waves as shown in Figures 6(c) and 6(d) (note that they are corresponding to the SLDs in Figures 6(e) and 6(f ), resp.). When the values of η are varying, the waves will move along the time axis (shown as the red lines). As a result, the combinations of the multiple time delays in the milling system will be altered, the stability nature will also be changed consequently.
In view of the significant influence of η on cutting stability, it is necessary to consider it for a VPCVSS process. However, as shown in Figure 2, it is obvious that the value of η is random and unmeasured in the actual machining process. Considering its periodic influence on the cutting stability region and the variation limit of the SLDs, one can take the median on the interval [η a , η b ] as the value of η for obtaining a relatively compromised SLD; thus, the difference between the predicting SLDs and the actual ones can be minimized. With this in mind, η can be defined as η a + (η a − η b )/2, that is, η � 3nπ/4 − π/2. us, some η, such as π/4 and 5π/4, are suggested and used in the practical application.

Influence of the Feed per Tooth.
Feed per tooth is an important basis for the selection of cutting parameters [40,41]. However, in most previous studies concerning milling chatter, it is ignored usually because it is considered to have no contribution to the dynamic cutting thickness and system stability. Also, one cannot get the effects of feed rate on the stability of milling by a linear cutting force model through the SMD in general. Here, the influence of feed per tooth on stability is investigated deeply, based on the method in this paper. Note that the following will be based on the nominal feed per tooth f m in equation (10) instead of the time-varying one f t,j (t, η) for convenience.
To illustrate the performance of feed per tooth on the dynamic behaviour of the VPCVSS system, SLDs are here provided for four different feed per tooth (i.e., f m � 0.01, 0.05, 0.1, and 0.2 mm/tooth) under two different radial immersion ratios (10% and 50%), as shown in Figures 4(a) and 4(b). It can be seen that, with the increasing of f m , the corresponding stability boundaries exhibit a trend of upward movement for both big and small radial depth of cut; thus, the limit depth of cut and stability area increases significantly. is is mainly due to the effect of the force-feed nonlinearity, specifically the nonlinear force coefficient q in equation (2), on chatter in the milling process. Because the value of q is equal to 0.75, that is, a number less than 1, a bigger f m will lead to a weaker nonlinear relationship between the cutting force and the dynamic displacements.
In order to further investigate the above effects, the SLDs corresponding to the four examples in Figure 4(a) are calculated under q � 0.94 and shown in Figure 4(c). It can be seen from Figure 4(c) that the influence of feed per tooth is the same as that of q � 0.75. However, compared with the latter, the former results in a more significant increase in stability. is is mainly due to the fact that since the values of q are significantly less than unity, the nonlinear feed effect is more evident. e conclusions are consistent with those in [41], where the machining stability of the nonlinear feed is investigated for general turning and milling process through experiments and theoretical analysis. us, the combination nature in the VPCVSS system does not change the general influence mechanism of the feed per tooth on milling stability.
Here, it must be noted that the above analysis conclusions are based on the premise that the value of q is less than one, which is suitable for some materials, such as aluminum alloy and steel [39]. However, if q � 1, the force-feed relationship in equations (2) and (15) will become linear and the effect of the force-feed nonlinearity on the system dynamic mentioned above will vanish. At this time, only if some other conditions are added, the influence of the feed effect can be considered again. For example, when the multiple time delays caused by the tool runout is considered, a smaller feed will result in a higher stability boundary [15]. For the case of the variable time delay caused by the tool path, [41] exhibits a shift of SLDs with the change of feed. Further, if q > 1, the associated conclusion in this article will be the opposite, that is, the increase of f m results in the smaller limit depth of cut and stability area. Obviously, different considerations and analysis angles will lead to changes in conclusions; this is understandable.
Next, the simplification of the time-varying characteristic of the feed per tooth is carried out and its influence on the stability is investigated. For equation (10), if the VSS effect is not considered, that is, RVA � 0, it will degenerate to if neither the VSS nor VPC effect is considered, it turns to Figure 7 shows the SLDs corresponding to f t,j (t, η) respectively defined by equations (10), (29), and (30). It can be seen that there is a small difference among their stability boundaries for both the radial immersion ratios of 10% and 50%. is indicates that no matter the multi-time-varying characteristics of feed per tooth reflected by equation (10) or the multiconstant ones by equation (29), their influence on the stability is very limited.
us, for the related general calculations, they both can be simplified to a single constant feed per tooth as shown in equation (30). However, this does not result in too much error for stability prediction.

3.4.
Influence of the Helix Angle. Although many efforts have been devoted to studying the influence of helix angle on stability in the past, the related works are almost restricted to the regular cutter [33,34] or variable pitch cutter [19]; there is little work focusing on this topic for VSS milling, let alone VPCVSS. In this section, the above question will be analyzed in detail through combining some new calculations with the ones in Figure 3.
As shown in Figures 3(a)-3(h), the stability limits increase evidently as the helix angle increases from β � 0°to 30°. And this trend is more significant for the small axial depth of cut. is is attributed to the fact that larger helix angle means more obvious system time variation in small immersion ratios, which is characterized as highly interrupted. Further, it can be found that, for VSS milling the effect of helix angle is mainly reflected in the high-speed region (e.g., among 12000-18000 rpm in Figures 3(a)-3(d)), whereas this effect is relatively small for the low-speed region. However, the effect of helix angle is very significant in both high-speed and low-speed regions for VPCVSS milling, for example, 2000-6000 rpm and 12000-18000 rpm in Figures 3(e)-3(h). is may be attributed to the combined effect of VPC and VSS, that is, the positive effect of VPC (especially linear pitches case) on the maximum depth of cut in low speed as shown in [17] and the effective suppression of period double chatter in the high-speed region for VSS milling as shown in [28].  In order to evaluate the stability trend caused by helix angle in more detail, the calculations are conducted under four different helix angles (i.e., β � 0°, 15°, 30°, and 45°) and their results are shown in Figure 8. It can be seen that, with the increase of the helix angle, the stable cutting area increases significantly. In Figure 8(a), the so-called "stable islands" [17,42] gradually appear around 2000 rpm so that the original unstable region turns to stable, whereas the sharp appendages around 16000 rpm gradually turn to "unstable islands" (flip bifurcation region) and disappear finally when β � 45°. However, it should be noted that, with the further increase of the helix angle, the influence of the helix angle on the cutting stability becomes weaker and weaker, especially in the high-speed area. is is mainly due to the smaller influence of the larger helix angle on the timevarying characteristics for the VPCVSS system.

Conclusion
is paper presents a dynamical model of the milling process of VPC and VSS considering the helix angle and the process phase difference. en, an updated SDM is proposed to gain the stability chart. After the verification of the proposed method by comparisons with previously published works and the time-domain simulations, lots of influence analyses are conducted deeply. From the stability analysis, the following conclusions can be drawn: (1) No matter for VSS or VPCVSS milling, the change of helix angle can result in significant stability difference for both small and big radial immersion ratios. However, for VSS milling, the effect of helix angle is mainly reflected in the high-speed region, whereas the effect of helix angle is very significant in both high-speed and low-speed regions for VPCVSS milling process. (2) Process phase difference has an important influence on the cutting stability, especially for the low-speed domain. Under the influence of the phase difference, there are two kinds of change limits in SLDs, which occur in η � (n − 1)π and nπ/2, n � 1, 2, .... erefore, it is necessary to consider the phase difference in the stability calculation of VPCVSS or similar system. Considering the randomness and immeasurability of η in the actual process, its recommended value is η � 3nπ/4 − π/2, such as π/4 and 5π/4. (3) Due to the effect of the force-feed nonlinearity, as the feed per tooth is increasing, the stability boundaries exhibit a trend of upward movement, both for big and small radial immersion ratios, thus the limit depths of cut and stability area are increasing significantly. (4) e multi-time-varying characteristic of feed per tooth in VPCVSS milling has almost negligible influence on the stability; thus, it can be simplified to a single constant one for a general calculation.

A:
State matrix for the complete system a p : Axial depth of cut (m) a D : Radial immersion ratio B j : State matrix for the system delays for the j-th tooth c x : Damping coefficient