A Parameter Optimization Method for Chatter Stability in Five-Axis Milling

Abstract

Chatter is a dynamic instability caused by the regeneration of waviness on the workpiece surface and damages the machining efficiency and product quality. For five-axis milling, the chatter stability analysis is more complicated because the cutter-workpiece engagement and the direction of the dynamic cutting force vary along the tool path. This paper presents a parameter optimization method to avoid chatter in the five-axis milling process. The dynamic milling system is modeled in each tool-path segment, and the stability is analyzed by introducing the semi-discretization method to solve the delay differential equation. In addition, the multi-frequency solution to the stability prediction for the multiple degrees of freedom spindle-cutter system and workpiece system is presented. Considering the speed and acceleration constraints of the spindle system and the feeding system, the spindle speed optimization iterative method is applied based on the chatter prediction. The verification experiment is conducted on the five-axis milling of the S-curve part to show the predictive accuracy of the chatter model and the validity of the proposed optimization method.

1. Introduction

Chatter is a kind of self-excited vibration formed in the machining process, which will cause a poor surface finish, increase tool wear rate, and reduce machining efficiency. Hence, researchers have carried out extensive studies on offline chatter prediction, online chatter detection, and chatter suppression [1]. In the study of chatter prediction, several analytical methods have been conducted to determine the stability lobe boundaries (SLD), such as the frequency-domain method, time-domain method, semi-discretization method, and full-discretization method. Altintas et al. [2,3] developed the frequency-domain method, which has been extended to the stability analysis for different particular milling processes, such as circular milling [4], low immersion milling [5], and five-axis ball-end milling [6]. Bayly et al. [7] improved a time-domain finite element analysis method for chatter prediction. The tool displacement during each element is approximated as a polynomial function, and a linear discrete dynamical version of the milling system is obtained [8]. T. Insperger et al. [9,10] introduced the semi-discretization method for the chatter stability analysis in the turning and the milling process. In addition, the stability of variable spindle speed milling is analyzed with varying delay terms established based on the semi-discretization method [11,12]. Gao et al. [13] improved the semi-discretization method for the chatter stability analysis in ultrasonic elliptical vibration-assisted milling. Y Ding et al. [14] presented a numerical solution to the chatter stability analysis by the full-discretization method. The chatter stability of the milling system is modeled and solved based on the numerical differentiation method [15]. The Floquet discriminant matrix of the delay differential equation (DDE) is constructed using the finite difference method and extrapolation method.

Besides different methods introduced to draw SLD, the influencing factors of SLD, such as cutting force coefficients, dynamic behavior, different ways of milling, etc., are widely researched by scholars. Pour and Torabizadeh [16] proposed a developed method for predicting SLD based on a time series analysis. The effect of the variable process damping was taken into consideration. Comak and Budak [17] developed a design method for the optimization of variable tool geometry to maximize chatter stability. Modeling dynamics and the stability of variable pitch and helix milling tools have been studied. Ozkirimli et al. [18] presented a generalized stability prediction approach for multi-axis milling with irregular cutting-edge geometries. Guo et al. [19] improved a more accurate SLD, considering the variation of cutting force coefficients. In flank milling of thin-walled parts, the chatter stability model considering force-induced deformation is presented by Li et al. [20,21]. Eto et al. [22] proposed a novel chatter-free milling strategy utilizing extraordinarily numerous flute endmill and high-speed machine tools. Higher material removal rate and stronger robustness can be achieved based on this method. Pelayo [23] presented an accurate dynamic milling force model for the circle-segment end mill. The effect of the main cutting factors (e.g., tool geometry, modal parameters, and cutting conditions) on the cutting force and the stability has been analyzed based on this model, which can be applied for the chatter detection in inclined operations.

For five-axis milling, the cutting condition, such as cutter-workpiece engagement (CWE) and the directional modal parameters, varies continuously with the change of feed direction and tool orientation. Hence, the chatter stability prediction of the five-axis milling process is more complicated compared with that of three-axis milling. In recent years, the study on chatter stability of five-axis milling has become a new research hotspot. Based on the frequency-domain method, the stability of five-axis milling has been predicted, and tool axis orientations in the chatter stability domain are iteratively searched at each location along the tool path [24]. Ma et al. [25] proposed a new analytical method of chatter stability, considering the change of the tool orientation for complex surface machining. Lu et al. [26] developed an improved numerical integration method to calculate the SLD of flat-end milling, and the influence of different tool postures was analyzed. Dai et al. [27] introduced an explicit precise time integration method to solve the DDE, and the CWE is extracted using a semi-analytical element method. J Li et al. [28] presented a chatter stability prediction method for the five-axis ball-end milling of the thin-walled workpiece. Zhang et al. [29,30] researched the effect of the variable pitch tool on the stability of five-axis ball-end milling and proposed an effective optimal pitch angle determination method.

Machining parameter optimization is one of the main methods to avoid chatter. Meanwhile, higher machining efficiency and surface quality can be realized by parameter adjustment according to the SLD. In recent decades, scholars have developed signficant research on machining parameter optimization with different optimization objectives, constraint conditions, and optimization algorithms [31,32,33]. Zhang and Ding [34] presented an augmented a Lagrangian function method for the optimization of machining parameters in the chatter-free milling process. Ringgaard et al. [35] designed a series of penalty cost functions to build a gradient-based bound optimization model for material removal rate maximization in the milling of thin-walled parts, and integrated the analytic hierarchy process and grey target decision methods for the parameter optimization, considering the material removal rate and surface location error. Meanwhile, different types of end mills have been designed and analyzed for machining operation to obtain a higher surface quality [36,37]. Sun et al. [38] presented a tool path generation method, considering the optimization of tool orientation, in five-axis milling with torus cutters. Meng et al. [39] proposed an optimal selection method of the barrel cutter with the aim of high productivity in the machining of blisk. Moreover, with the development of artificial intelligent algorithms, some researchers have introduced these optimization methods to machining parameter optimization [40,41]. Mishra et al. [42] developed a multi-objective particle swarm machining parameters optimization to maximize the metal removal rate and minimize chatter. A genetic algorithm was applied to the process parameter optimization model in the face milling, considering the constraint of the chatter stability domain [43]. Mokhtari et al. [44] presented a parameter optimization procedure based on the genetic algorithm to maximize the allowable cutting depth for the chatter-free milling process. Wang et al. [45] employed the genetic algorithm to optimize the parameters of the spindle speed variation to avoid chatter effectively. However, the above research mainly focuses on the parameter optimization method for three-axis chatter-free milling. In this paper, a novel rolling iterative parameter optimization method with the constraints of the chatter-free condition, spindle speed, and acceleration for five-axis milling is proposed. Meanwhile, the semi-discretion method is developed for the chatter analysis in five-axis milling, and the multi-frequency solution is derived.

In Section 2, the procedure of the chatter-free parameter optimization approach is presented. In Section 3, the dynamic milling force and kinematic transformation for five-axis milling are modeled. In Section 4, chatter mechanism analysis based on the semi-discretization method is discussed. In Section 5, the chatter prediction procedure and parameter optimization algorithm are presented. Furthermore, the proposed optimization method is experimentally proven in the five-axis milling test in Section 6. Finally, the conclusion is presented in Section 7.

2. The Chatter-Free Parameter Optimization Approach in Five-Axis Milling

Figure 1 illustrates the procedure of the chatter-free parameter optimization approach. Above all, the G-code file, the workpiece model, and the cutter geometry information are loaded into the system for the geometric modeling. Based on the five-axis kinematic transformation, the tool point position (x, y, z) and the tool orientation (i, j, k) in the workpiece coordinate system (WCS) are calculated at the discrete location in the segment of the tool path. Furthermore, the tool swept volume is created based on the two adjacent tool positions. The CWE is extracted by the Boolean intersection operation between the current tool profile surface and the solid model of the workpiece. Then, the chatter stability prediction model for five-axis milling is built based on the CWE calculation, the coordinate transformation matrix, and the identified parameters (e.g., modal parameters and cutting coefficients). Therefore, the stability is predicted by applying the semi-discretization to solve the DDE. Next, chatter-free parameter optimization is carried out with the constraints of spindle speed, spindle acceleration, et al. If all the parameters satisfy the constraints, the optimized cutting parameters are rewritten in the G-code file. If a constraint is not met, the machining parameters are selected around the original set or recursively searched for optimization. Finally, the optimized G-code file is finished for the chatter-free five-axis machining.

Compared with the traditional SLD-based parameter selection method, the change of cutting condition is considered in this proposed approach, which makes it more accurate in five-axis milling. In addition, this chatter-free parameter optimization integrated with the virtual machining simulation has these three following advantages:

• Offline simulation ensures sufficient calculation time;
• Predicting optimization with constraints makes the machining process more operable and more reliable;
• This rolling optimization strategy takes the machining condition along the whole tool path into consideration.

3. Dynamic Milling Force and Kinematic Transformation

3.1. Dynamic Milling Force Model

In this section, the dynamic milling forces are calculated based on the instantaneous mechanistic force model [46]. The tangential differential force dF_tjk, radial differential force dF_rjk, and axial differential force dF_ajk acting on the kth cutting element of the jth tooth are represented as

$$\begin{array}{l}d{F}_{tjk}=K_{tc}h\left({\varphi }_{jk}\right)db+K_{te}dS\\ d{F}_{rjk}=K_{rc}h\left({\varphi }_{jk}\right)db+K_{re}dS\\ d{F}_{ajk}=K_{ac}h\left({\varphi }_{jk}\right)db+K_{ae}dS\end{array}$$

(1)

where ϕ_jk is the instantaneous immersion angle and h(ϕ_jk) is the undeformed chip thickness. K_tc, K_rc, K_ac, K_te, K_re, and K_ae are the cutting force coefficients, db is chip width, and dS is the cutter segment length.

The expression for the chip thickness at height z of the jth flute is given by

$$h\left({\varphi }_{jk},z\right)=\epsilon \left({\varphi }_{jk}\left(z\right)\right)\left[c_{Xj}\left(z\right)\sin \left({\varphi }_{jk}\left(z\right)-{\theta }_s\left(z\right)\right)\sin \left(\kappa \left(z\right)\right)-c_{Zj}\left(z\right)\cos \left(\kappa \left(z\right)\right)\right]$$

(2)

where c_Xj and c_Zj are the feed per flute in the horizontal direction and the vertical direction, respectively. κ is the axial immersion angle and θ_s is the shift angle. ε(ϕ_jk(z)) is a criterion function to determine whether the tooth is in cut based on the entry (ϕ_st) and exit (ϕ_ex) immersion angle pairs. Hence, if the cutting edge is engaged with the workpiece, ε equals 1, and else, ε equals 0.

The dynamic chip thickness is calculated from Equation (2), which can be expressed as

$$\begin{array}{c}{h}_d\left({\varphi }_{jk},z\right)=\epsilon \underset{n\left({\varphi }_{jk}\left(z\right),z\right)}{\underbrace{\left[\begin{array}{ccc}\sin {\varphi }_{jk}\left(z\right)\sin \left(\kappa \left(z\right)\right)& \cos {\varphi }_{jk}\left(z\right)\sin \left(\kappa \left(z\right)\right)& \cos \left(\kappa \left(z\right)\right)\end{array}\right]}}{\left\{\begin{array}{c}{\Delta }_X\left(t\right)\\ {\Delta }_Y\left(t\right)\\ {\Delta }_Z\left(t\right)\end{array}\right\}}_{FCS}\\ =\epsilon \left({\varphi }_{jk}\left(z\right)\right)n\left({\varphi }_{jk}\left(z\right),z\right){\left\{\begin{array}{c}{\Delta }_X\left(t\right)\\ {\Delta }_Y\left(t\right)\\ {\Delta }_Z\left(t\right)\end{array}\right\}}_{FCS}\end{array}$$

(3)

where the displacement difference vector ${\left\{\begin{array}{ccc}{\Delta }_X\left(t\right)& {\Delta }_Y\left(t\right)& {\Delta }_Z\left(t\right)\end{array}\right\}}_{FCS}^T$ is defined as the difference between the displacement ${\left\{\begin{array}{ccc}x\left(t\right)& y\left(t\right)& z\left(t\right)\end{array}\right\}}_{FCS}^T$ at present and the displacement ${\left\{\begin{array}{ccc}x\left(t-\tau \right)& y\left(t-\tau \right)& z\left(t-\tau \right)\end{array}\right\}}_{FCS}^T$ one tooth period before in the feed coordinate system (FCS). τ is the one tooth period and equals τ = 60/(Nn), where N is the num of flutes, and n is the spindle speed. Moreover, the following equation holds:

$${\left\{\begin{array}{c}{\Delta }_X\left(t\right)\\ {\Delta }_Y\left(t\right)\\ {\Delta }_Z\left(t\right)\end{array}\right\}}_{FCS}={\left\{\begin{array}{c}x\left(t\right)\\ y\left(t\right)\\ z\left(t\right)\end{array}\right\}}_{FCS}-{\left\{\begin{array}{c}x\left(t-\tau \right)\\ y\left(t-\tau \right)\\ z\left(t-\tau \right)\end{array}\right\}}_{FCS}$$

(4)

By inserting Equation (3) to Equation (1), the dynamic cutting force can be derived as follows:

$$\begin{array}{c}{\left\{\begin{array}{c}{F}_{x,d}\\ F_{y,d}\\ F_{z,d}\end{array}\right\}}_{FCS}={\displaystyle \sum _{j=1}^N{\displaystyle \sum _{k=1}^M\left(H\left\{\begin{array}{c}{K}_{tc}\\ K_{rc}\\ K_{ac}\end{array}\right\}\epsilon \left({\varphi }_{jk}\left(z\right)\right)dbn\left({\varphi }_{jk}\left(z\right),z\right){\left\{\begin{array}{c}{\Delta }_X\left(t\right)\\ {\Delta }_Y\left(t\right)\\ {\Delta }_Z\left(t\right)\end{array}\right\}}_{FCS}\right)}}\\ =\underset{\Psi \left(\varphi \right)}{\underbrace{{\displaystyle \sum _{j=1}^N{\displaystyle \sum _{k=1}^M\left(H\left\{\begin{array}{c}{K}_{tc}\\ K_{rc}\\ K_{ac}\end{array}\right\}\epsilon \left({\varphi }_{jk}\left(z\right)\right)dbn\left({\varphi }_{jk}\left(z\right),z\right)\right)}}}}{\left\{\begin{array}{c}{\Delta }_X\left(t\right)\\ {\Delta }_Y\left(t\right)\\ {\Delta }_Z\left(t\right)\end{array}\right\}}_{FCS}\end{array}$$

(5)

where H is the transformation matrix and equals

$$H=\left[\begin{array}{ccc}-\cos \left({\varphi }_{jk}\left(z\right)\right)& -\sin \left(\kappa \left(z\right)\right)\sin \left({\varphi }_{jk}\left(z\right)\right)& -\cos \left(\kappa \left(z\right)\right)\sin \left({\varphi }_{jk}\left(z\right)\right)\\ \sin \left({\varphi }_{jk}\left(z\right)\right)& -\sin \left(\kappa \left(z\right)\right)\cos \left({\varphi }_{jk}\left(z\right)\right)& -\cos \left(\kappa \left(z\right)\right)\cos \left({\varphi }_{jk}\left(z\right)\right)\\ 0& \cos \left(\kappa \left(z\right)\right)& -\sin \left(\kappa \left(z\right)\right)\end{array}\right]$$

(6)

Via the coordinate transformation, the cutting forces on the jth flute at height z are transformed from tangential, radial, and axial directions to the x-y-z directions in FCS.

3.2. Transformations to the Machine Coordinate

For five-axis milling, the feed direction and tool orientation change along the tool path. The modal parameters of the spindle-cutter system are measured by impact testing. Hence, the transformation from FCS to machine coordinate system (MCS) is constructed for the stability analysis.

As shown in Figure 2, the cutting forces transform from FCS to WCS as

$${\left\{\begin{array}{c}{F}_x\\ F_y\\ F_z\end{array}\right\}}_{WCS}=R_{FW}{\left\{\begin{array}{c}{F}_x\\ F_y\\ F_z\end{array}\right\}}_{FCS}$$

(7)

where R_FW is the transformation matrix, and equals R_FW = [{X_T} {Y_T} {Z_T}]. {X_T}, {Y_T}, and {Z_T} are the unite x-axis, y-axis, and z-axis direction vectors of FCS expressed in WCS, respectively, which can be calculated with the cutter location (CL) data [47]. {P_i} and {T_i} are the tool tip coordinate vector {x, y, z}_wcs and tool orientation vector {i, j, k}_wcs, respectively, at the ith tool path segment. The feed direction vector {F_T} is given as

$$\left\{F_T\right\}=\frac{\left\{P_{i+1}\right\}-\left\{P_i\right\}}{\left|\left\{P_{i+1}\right\}-\left\{P_i\right\}\right|}$$

(8)

The {X_T}, {Y_T}, and {Z_T} are calculated as

$$\left\{Z_T\right\}=\frac{\left\{T_{i+1}\right\}}{\left|\left\{T_{i+1}\right\}\right|}, \left\{Y_T\right\}=\frac{\left\{T_{i+1}\right\}\times \left\{F_T\right\}}{\left|\left\{T_{i+1}\right\}\times \left\{F_T\right\}\right|}, \left\{X_T\right\}=\frac{\left\{Y_T\right\}\times \left\{Z_T\right\}}{\left|\left\{Y_T\right\}\times \left\{Z_T\right\}\right|}$$

(9)

CL data can be obtained from each axis position. In addition, for an AC-type five-axis machine, {P_i} = {x, y, z} and {T_i} = {i, j, k} are calculated with the positions (x_t, y_t, z_t, a_t, c_t) through the kinematic transformation:

$$\{\begin{array}{l}x=x_t\cos \left(c_t\right)-y_t\cos \left(a_t\right)\sin \left(c_t\right)+z_t\sin \left(a_t\right)\sin \left(c_t\right)\hfill \\ y=x_{t}sin\left(c_t\right)+y_t\cos \left(a_t\right)\cos \left(c_t\right)-z_t\sin \left(a_t\right)\cos \left(c_t\right)\hfill \\ z=y_t\sin \left(a_t\right)+z_t\cos \left(a_t\right)\hfill \\ i=\sin \left(a_t\right)\sin \left(c_t\right)\hfill \\ j=-\sin \left(a_t\right)\cos \left(c_t\right)\hfill \\ k=\cos \left(a_t\right)\hfill \end{array}$$

(10)

Therefore, the rotational transfer matrix from WCS to MCS is formulated as follows:

$$R_{WM}=R_z\left(c_t\right)R_X\left(a_t\right)=\left[\begin{array}{ccc}\cos \left(c_t\right)& -\sin \left(c_t\right)\cos \left(a_t\right)& \sin \left(c_t\right)\sin \left(a_t\right)\\ \sin \left(c_t\right)& \cos \left(c_t\right)\cos \left(a_t\right)& -\cos \left(c_t\right)\sin \left(a_t\right)\\ 0& \sin \left(a_t\right)& \cos \left(a_t\right)\end{array}\right]$$

(11)

Hence, the dynamic cutting forces expressed in the MCS are as follows:

$${\left\{\begin{array}{c}{F}_{x,d}\\ F_{y,d}\\ F_{z,d}\end{array}\right\}}_{MCS}=R_{WM}{\left\{\begin{array}{c}{F}_{x,d}\\ F_{y,d}\\ F_{z,d}\end{array}\right\}}_{WCS}$$

(12)

The dynamic displacement transforms from FCS to MCS, which is given as follows:

$${\left\{\begin{array}{c}{\Delta }_X\left(t\right)\\ {\Delta }_Y\left(t\right)\\ {\Delta }_Z\left(t\right)\end{array}\right\}}_{FCS}=R_{FW}^{-1}{\left\{\begin{array}{c}{\Delta }_X\left(t\right)\\ {\Delta }_Y\left(t\right)\\ {\Delta }_Z\left(t\right)\end{array}\right\}}_{WCS}=R_{FW}^{-1}{R}_{WM}^{-1}{\left\{\begin{array}{c}{\Delta }_X\left(t\right)\\ {\Delta }_Y\left(t\right)\\ {\Delta }_Z\left(t\right)\end{array}\right\}}_{MCS}$$

(13)

By substituting Equations (5), (7), and (13) into Equation (12), the dynamic cutting forces in the MCS are derived:

$${\left\{\begin{array}{c}{F}_{x,d}\\ F_{y,d}\\ F_{z,d}\end{array}\right\}}_{MCS}=R_{WM}{R}_{FW}{\left\{\begin{array}{c}{F}_{x,d}\\ F_{y,d}\\ F_{z,d}\end{array}\right\}}_{FCS}=R_{FM}\Psi \left(\varphi \right)R_{FM}^{-1}{\left\{\begin{array}{c}{\Delta }_X\left(t\right)\\ {\Delta }_Y\left(t\right)\\ {\Delta }_Z\left(t\right)\end{array}\right\}}_{MCS}$$

(14)

where R_FM is the matrix for the transformation from FCS to MCS and equals $R_{FW}=R_{WM}{R}_{FW}$.

4. Chatter Stability Prediction for Five-Axis Milling

4.1. Chatter Stability Prediction Model

While the stiffness of the spindle-cutter system in the z-direction is assumed to be rigid, the dynamic milling system is simplified as a basic two-degrees-of-freedom system in the MCS, as shown in Figure 3. The dynamic equations can be expressed as:

$$\left[\begin{array}{cc}{m}_x& 0\\ 0& m_y\end{array}\right]\left[\begin{array}{c}\ddot{x}\left(t\right)\\ \ddot{y}\left(t\right)\end{array}\right]+\left[\begin{array}{cc}{c}_x& 0\\ 0& c_y\end{array}\right]\left[\begin{array}{c}\dot{x}\left(t\right)\\ \dot{y}\left(t\right)\end{array}\right]+\left[\begin{array}{cc}{k}_x& 0\\ 0& k_y\end{array}\right]\left[\begin{array}{c}x\left(t\right)\\ y\left(t\right)\end{array}\right]=\left[\begin{array}{c}{F}_{x,d}\left(t\right)\\ F_{y,d}\left(t\right)\end{array}\right]$$

(15)

By substituting Equation (14) into Equation (15), Equation (15) can be rewritten as follows:

$$M\ddot{\zeta }\left(t\right)+C\dot{\zeta }\left(t\right)+K\zeta \left(t\right)=W(t)\left\{\zeta \left(t\right)-\zeta \left(t-\tau \right)\right\}$$

(16)

where M, C, and K denote the system mass, damping, and stiffness matrices, respectively. ζ(t) represents the displacement vector in the x- and y-directions of MCS and equals ${\left\{\begin{array}{cc}x\left(t\right)& y\left(t\right)\end{array}\right\}}_{MCS}^T$. W(t) is the cutting force coefficient submatrix derived from Equation (14), which consists of the elements in the first two columns and the first two rows of $R_{FM}\Psi \left(t\right)R_{FM}^{-1}$:

$$R_{FM}\Psi \left(t\right)R_{FM}^{-1}=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]W\left(t\right)=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]$$

(17)

Equation (16) can be rewritten as the space state equation in the time domain:

$$\begin{array}{l}\dot{q}\left(t\right)=A\left(t\right)q\left(t\right)+B\left(t\right)u\left(t-\tau \right)\\ u\left(t\right)=Dq\left(t\right)\end{array}$$

(18)

where

$$q\left(t\right)=\left(\begin{array}{c}\zeta \left(t\right)\\ \dot{\zeta }\left(t\right)\end{array}\right)$$

(19)

$$A\left(t\right)=\left(\begin{array}{cc}0& I\\ -\left(M^{-1}\left(K-W\left(t\right)\right)\right)& -M^{-1}C\end{array}\right)B\left(t\right)=\left(\begin{array}{c}0\\ -M^{-1}W\left(t\right)\end{array}\right)D=\left(\begin{array}{cc}I& 0\end{array}\right)$$

(20)

I is the identity matrix (with 2 × 2 dimensions). Based on the semi-discretization method [48], the DDE Equation (18) is approximated based on a sequence of the ordinary differential equation (ODE) with discrete time intervals. Let h_T be the sampling period of the path segment, so that τ = ph_T, and $p=\mathrm{int}\left(\frac{\tau }{h_T}+\frac{1}{2}\right)$. In the ith discrete interval [t_i, t_i+1], the semi-discrete system of Equation (18) is described as

$$q_{i+1}=e^{A_i{h}_T}{q}_i+\frac{1}{2}\left(e^{A_i{h}_T}-I\right)A_i{}^{-1}{B}_i\left(q_{i-p}+q_{i-p+1}\right)$$

(21)

Equations (21) and (18) can be rewritten as the discrete map form:

$$z_{i+1}=Q_i{z}_i$$

(22)

where z_i and z_i+1 represent the augmented state vectors in the ith and (i + 1)th discrete interval:

$$z_i={\left(\begin{array}{cccc}{q}_i& u_{i-1}& \dots & u_{i-p}\end{array}\right)}^T={\left(\begin{array}{cccccc}{\zeta }_i& {\dot{\zeta }}_i& {\zeta }_{i-1}& {\zeta }_{i-2}& \dots & {\zeta }_{i-p}\end{array}\right)}^T$$

(23)

Moreover, Q_i denotes the transition matrix, which can be derived as

$$Q_i=\left(\begin{array}{cccccc}{e}^{A_i{h}_T}& 0& \cdots & 0& \frac{1}{2}\left(e^{A_i{h}_T}-I\right)A_i{}^{-1}{B}_i& \frac{1}{2}\left(e^{A_i{h}_T}-I\right)A_i{}^{-1}{B}_i\\ D& 0& \cdots & 0& 0& 0\\ 0& I& \cdots & 0& 0& 0\\ ⋮& & & & & ⋮\\ 0& 0& \cdots & 0& I& 0\end{array}\right)$$

(24)

Thus, the transition equation of the states is given by coupling the coefficient matrices in each interval within a simulation period τ:

$$z_r=Q_r\cdots Q_2{Q}_1{z}_0=Q{z}_0$$

(25)

Based on the Floquet theory, the eigenvalues of the transition matrix Q are used to judge the stability of the system. If all modulus of the eigenvalues of Q(t) are less than one, the system is asymptotically stable.

4.2. Multi-Frequency Analysis for Chatter Stability

In Equation (15), the dynamic milling system is simplified as two single-degrees-of-freedom (SDOF) systems in the x- and y-directions. However, the machine tool has multiple degrees of freedom (MDOF) in different directions. The transfer function in each direction can be expressed as the accumulation of second-order differential equations in the Laplace domain:

$$h(s)={{\displaystyle \sum _{k=1}^n\left[\frac{{\alpha }_k+{\beta }_{k}s}{s^2+2{\xi }_k{\omega }_{n,k}s+{\omega }_{n,k}^2}\right]}}_{mod{e}^{}k}$$

(26)

where α_k and β_k are the real parts and the imaginary parts of the numerator in kth mode, respectively. Considering the real mode shape, the imaginary part of the numerator is negligible (β_k = 0). The transfer function is given as

$$\frac{X\left(s\right)}{F\left(s\right)}={\displaystyle \sum _{k=1}^n{u}_k^2\frac{1}{m_k{s}^2+c_{k}s+k_k}}$$

(27)

where u_k is the gain of the real mode shape. The dynamic displacements in each direction are decomposed into the modal displacement components. Therefore, the dynamic equations (Equation (15)) are modified as

$$M_q{\ddot{x}}_q+C_q{\dot{x}}_q+K_q{x}_q=u_q{F}_{q,d} (q=\mathrm{x}, \mathrm{y})$$

(28)

where the subscript q represents the different directions, and M_q, C_q, and K_q are the mass, damping, and stiffness matrices, respectively. u_q is the vector of mode shape:

$$u_q=\left[\begin{array}{ccc}{u}_{q,1}^2& \dots & u_{q,k}^2\end{array}\right]$$

(29)

Equation (16) can be rewritten as follows for MDOF:

Where $M_t=\left[\begin{array}{cc}{M}_x& \\ & M_y\end{array}\right]$, $C_t=\left[\begin{array}{cc}{C}_x& \\ & C_y\end{array}\right]$, $K_t=\left[\begin{array}{cc}{K}_x& \\ & K_y\end{array}\right]$, $E=\left[\begin{array}{cc}\stackrel{k_x}{\overbrace{\begin{array}{ccc}1& \dots & 1\end{array}}}& \\ & \underset{k_y}{\underbrace{\begin{array}{ccc}1& \cdots & 1\end{array}}}\end{array}\right]$, $U={\left[\begin{array}{cc}\begin{array}{ccc}{u}^2{}_{x,1}& \dots & u^2{}_{x,k_x}\end{array}& \\ & \begin{array}{ccc}{u}^2{}_{y,1}& \cdots & u^2{}_{y,k_y}\end{array}\end{array}\right]}^T$, ${\zeta }_t\left(t\right)={\left[\begin{array}{cccccc}{x}_{t,1}& \cdot \cdot \cdot & x_{t,kx}& y_{t,1}& \cdot \cdot \cdot & y_{t,ky}\end{array}\right]}^T$.

Hence, the solution of Equation (28) for MDOF is the same as Equation (16) for SDOF with $W\left(t\right)=U{R}_{FM}\Psi \left(t\right)R_{FM}^{-1}E$.

4.3. The Coupling Effect of the Workpiece Stiffness

For weak stiffness workpiece machining, the vibration of the workpiece should be taken into consideration. The dynamic equations of the coupling system are established as follows:

$$\begin{array}{l}{M}_w{\ddot{\zeta }}_w\left(t\right)+C_w{\dot{\zeta }}_w\left(t\right)+K_w{\zeta }_w\left(t\right)=F_{d,FCS}\\ M_t{\ddot{\zeta }}_t\left(t\right)+C_t{\dot{\zeta }}_t\left(t\right)+K_t{\zeta }_t\left(t\right)=F_{d,MCS}\\ {\Delta }_{FCS}={\Delta }_{w,FCS}+{\Delta }_{t,FCS}\end{array}$$

(30)

where M_w, C_w, and K_w denote the mass, damping, and stiffness matrices for the workpiece, respectively. ζ_t(t) is the model displacement. Hence, the modification of Equation (29) is presented as

$$M\ddot{\zeta }\left(t\right)+C\dot{\zeta }\left(t\right)+K\zeta \left(t\right)=\left[\begin{array}{c}U{R}_{FM}\\ I\end{array}\right]\Psi \left(\varphi \left(t\right)\right)\left[\begin{array}{cc}{R}_{FM}^{-1}E& I\end{array}\right]\left\{\zeta \left(t\right)-\zeta \left(t-T\right)\right\}$$

(31)

where $M=\left[\begin{array}{cc}{{\rm M}}_t& \\ & {{\rm M}}_w\end{array}\right]$, $C=\left[\begin{array}{cc}{C}_t& \\ & C_w\end{array}\right]$, $K=\left[\begin{array}{cc}{K}_t& \\ & K_w\end{array}\right]$, $\zeta \left(t\right)={\left[\begin{array}{cccccc}\begin{array}{ccc}{x}_{t,1}& \cdot \cdot \cdot & x_{t,kx}\end{array}& \begin{array}{ccc}{y}_{t,1}& \cdot \cdot \cdot & y_{t,ky}\end{array}& z_t& x_w& y_w& z_w\end{array}\right]}^T$.

Considering the stiffness of the spindle-cutter system in the z-direction and the stiffness of the workpiece in the feed and axial direction are rigid, the system is simplified as follows:

$$\left[\begin{array}{c}U{R}_{FM}\\ I\end{array}\right]\Psi \left(t\right)\left[\begin{array}{cc}{R}_{FM}^{-1}E& I\end{array}\right]=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdot \cdot \cdot & a_{16}\\ a_{21}& a_{22}& \cdot \cdot \cdot & a_{26}\\ ⋮& ⋮& \ddots & ⋮\\ a_{61}& a_{62}& \cdot \cdot \cdot & a_{66}\end{array}\right]W\left(t\right)=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{16}\\ a_{21}& a_{22}& \dots & a_{26}\\ a_{51}& a_{52}& \dots & a_{56}\end{array}\right]$$

(32)

where $\zeta \left(t\right)={\left[\begin{array}{ccccccc}{x}_{t,1}& \cdot \cdot \cdot & x_{t,kx}& y_{t,1}& \cdot \cdot \cdot & y_{t,ky}& y_w\end{array}\right]}^T$.

In this study, an S-curve thin-walled part five-axis milling is carried out as the machining experiment. The operation of the hammer impact test in the particular position (e.g., the center of the S-curve and the inner corner point) of the workpiece is inconvenient. For the workpiece model analysis, the contact constraint can be regarded as rigidity so that the theoretical calculation has good accuracy. Therefore, the S-curve thin-walled specimen is modeled as a cantilever beam for modal analysis. According to classical bending theory, the natural frequency ω_j and modal shape ψ_j are present:

$${\omega }_j={\left(\frac{EI}{\rho A{L}^4}\right)}^{1/2}{a}_j^2$$

(33)

$$\psi \left(x\right)=C_1\left\{\left[\sin \left({\alpha }_j\frac{x}{L}\right)-\sinh \left({\alpha }_j\frac{x}{L}\right)\right]-\frac{\sin \left({\alpha }_j\right)+\sinh \left({\alpha }_j\right)}{\cos \left({\alpha }_j\right)+\cosh \left({\alpha }_j\right)}\left[\cos \left({\alpha }_j\frac{x}{L}\right)-\cosh \left({\alpha }_j\frac{x}{L}\right)\right]\right\}$$

(34)

where L is the length of the beam, A is the sectional area, E is the Young modulus, ρ is the density, I is the inertia, x is the position along the beam, and α_j is the root of the cantilever bending characteristic equation:

$$\cos \left(\alpha \right)\cosh \left(\alpha \right)+1=0$$

(35)

The S-curve is segmented into rectangular sections and loop sections. For the rectangular section, the inertia I_xr equals

$$I_{xy}=\frac{1}{12}w{h}^3$$

(36)

where w and h are the rectangular section’s width and height, respectively. For the loop section, the inertia I_xl equals

$$I_{xl}=\frac{1}{8}\left(R^4-r^4\right)\theta +\frac{1}{2}{\left(\frac{R+r}{2}\right)}^2\theta +\frac{1}{8}\left(R^4-r^4\right)\sin \theta -\frac{4}{3}\left(\frac{R+r}{2}\right)\left(R^3-r^3\right)\sin \frac{\theta }{2}$$

(37)

where r and R are radiuses of the inner and outer circle, respectively, and θ is the intersection angle of the loop. Hence, the natural frequency of the S-curve part workpiece at the discrete location can be calculated by Equation (33), as presented in Figure 4.

However, the accuracy of the proposed chatter prediction method can be increased by building a more accurate workpiece model through the finite element method, especially for the thin-walled workpiece with the sculptured surface [49].

5. Parameter Optimization for Chatter Stability

For the five-axis milling process, the geometric condition constantly changes along the tool path with the variation of the feed direction and the tool orientation, as shown in Figure 2 and Figure 5. The engagement angles (ϕ_st, ϕ_ex) are identified by the CWE, which is obtained using the machining simulation system developed by the author’s laboratory. Based on the solid modeling method [50,51], the CWE map is extracted through the intersection Boolean operation between the solid model of the tool and the workpiece at each discrete interval. As shown in Figure 5, the envelope of the tool model and the contact patch between the workpiece and the tool are projected to the y-z plane in FCS. In addition, the engagement angles are calculated at each discrete height along the tool orientation.

Because

$$l_{path}=hf p=\frac{60}{nNh}=\frac{60}{l_{path}}\frac{f}{Nn}$$

(38)

where l_path is the length of the path segment, if p and l_path remain constant, the feed rate f varies in direct proportion to the spindle speed n.

Therefore, when unstable conditions are predicted, the stable speed is searched around the current spindle speed n for optimization. By the stability prediction of five-axis milling, the simulation procedure and the parameters selection algorithm are given in Figure 6 and Figure 7, where n_max and n_min are the maximum and minimum spindle speed constraints to ensure the machine works under stable machining conditions. ω_c is the chatter frequency, which can be calculated by Equation (39):

$${\omega }_c=\mathrm{Im}(\ln (\mu ))/\tau$$

(39)

Δn is the speed step size in each iteration, α_max is the maximum spindle acceleration, opti_i is the iteration count in the inner loop considering the acceleration and deceleration of the spindle speed limit, and α_{opt_i} is the calculated spindle acceleration according to the optimized speed. If the spindle acceleration is over the limit of acceleration constraints, the spindle speed in the previous segment should be modified as follows:

$$n(i-opti\_i)=\frac{1}{2}\sqrt{n_{ki}^2-2{\alpha }_{\max }{n}_{i-opti\_i}(t_{i-opti\_i}-t_{i-opti\_i-1})}+\frac{1}{2}{n}_{ki}$$

(40)

The spindle speed is rollback-optimized, and the transition matrix Q_i is calculated accordingly. In addition, k_max is the maximum iteration. If the system has exceeded the maximum stability threshold, the search for optimum stable spindle speed can terminate by introducing k_max.

6. Simulation and Experimental Verification

The S-curve machining test, as an international standard, is usually used to verify the five-axis machining performance [52]. As shown in Figure 8, an S-curve part with sculptured surfaces is finishing-machined to verify the effectiveness of the proposed chatter prediction model and parameter optimization method. A five-axis machine tool XH7132A and a two-fluted, 8 mm diameter ball-end mill with a 30-degree helix is used for the five-axis machining test. The frequency response functions (FRFs) of the spindle-cutter system are measured by applying the hammering tests to the ball-end mill in the x- and y-directions of the MCS. The identified modal parameters from the measured and modified FRFs (Figure 9) are presented in

Table 1. Model parameters of the spindle-cutter system.

Mode Direction	Mode	Natural Frequency fn(Hz)	Damping Ratio ξ	Stiffness k (N/m)
x	1	1602	0.02311	8.4395 × 106
	2	2631	0.06811	2.4281 × 107
	3	3413	0.04044	4.3693 × 107
	4	5331	0.02848	2.7526 × 106
y	1	1470	0.01519	2.4049 × 107
	2	1632	0.01162	4.4599 × 107
	3	2804	0.04717	5.1011 × 107
	4	3878	0.03086	1.3455 × 107
	5	5116	0.02083	5.0825 × 106
	6	5490	0.01179	2.2921 × 107

. The cutting force coefficients of the workpiece material Aluminum 7075, as presented in

Table 2. Cutting force coefficients of Aluminum 7075.

Kt (N/mm2)	Kr (N/mm2)	Ka (N/mm2)
796.1	168.8	222.0

, are identified from the cutting force test using a table dynamometer Kistler 9257B. The parameters of the workpiece material and cross-section are presented in

Table 3. Parameters of workpiece material and cross-section.

Density ρ (kg/m3)	Modulus E (N/m2)	Length L (mm)	Height h (mm)
2700	6.89 × 1010	16	2.2

. The simulation parameters for optimization are presented in

Table 4. Simulation parameters for optimization.

Parameters	Value
Step of sampling segment p	60
Discrete axial cut depth dz	0.5 mm
Discrete immersion angle dϕ	6 degrees
Maximum iteration kmax	10
Speed step size Δn	200 r/min
Maximum spindle acceleration αmax	1800 r/(min/s)
Minimum spindle speed nmin	2000 r/min
Maximum spindle speed nmax	8000 r/min

. The finish machining of this S-curve part has a 1 mm allowance in the normal direction of the surface. The engagement geometry and radius of the loop cross-section are identified by a self-developed machining simulation system, as shown in Figure 10. This machining simulation software is developed based on the C++ programming language with the development platform of Visual Studio 2019 and the open architecture of Qt. Meanwhile, the three-dimensional engine AnyCAD is used for geometric modeling and scene rendering. This virtual machining system has been used for the digital twin building with the function of material removal processing simulation, dynamic cutting force simulation, and contour error analysis [53].

Before the optimization, the spindle speed is constant at 5000 r/min, and the feed velocity is constant at 900 mm/min. The positions of each axis are presented in Figure 11. The CWE at the different discrete locations is shown in Figure 12 and Figure 13(a)–(f) to illustrate the chip section evolution.

The chatter prediction result is shown in Figure 14, with the maximum eigenvalues of the transition matrix at each tool position along the tool path. As discussed in Section 3.1, the eigenvalue should be less than one for a stable dynamic system. As shown in Figure 14b, the tool locations with eigenvalues greater than one are marked in red. Based on the optimization algorithm proposed in Section 4, the spindle speed is optimized, as shown in Figure 15. The chatter prediction result for the optimized spindle speed is shown in Figure 16. It can be found that all eigenvalues in each position are less than one.

The cutting test results before and after the parameter optimization are shown in Figure 17 and Figure 18. By comparison of the surface quality and the sound spectrum, the cutting chatter is detected at the beginning and the end of the S-curve, which is caused by the weak stiffness of the workpiece, as presented in Figure 4. For stable machining, the major frequency should be the multiplier of the tooth passing frequency. Conversely, the chatter frequency will stick out in the frequency components when the chatter occurs. From the frequency analysis results presented in Figure 18, the dominant frequency of 728 Hz is closer to the chatter frequency (731 Hz) calculated by Equation (39) in the milling process. Meanwhile, this chatter frequency is from the coupling effect of the workpiece and the tool, considering the frequency of 728 Hz is not clear to the natural frequency of the workpiece or the tool either. After the optimization, the cutting process is stable, and the major frequencies of the sound spectrum are concentrated on the tooth passing frequency (200 Hz) multiplication. As shown in Figure 14b, the chatter phenomenon gets suppressed with the stiffness of the workpiece enhanced along the tool path. Besides, it can be observed that chatter marks occur in the inner corner of the S-curve, as shown in Figure 17. The main reason is the increase of the cutting engagement geometry (Figure 13e), which is in agreement with the prediction in Figure 14. When the spindle speed is optimized by the method presented in Figure 7, the chatter is eliminated, and the surface finish becomes smooth.

7. Conclusions

This study proposed a parameters optimization method for chatter stability in the five-axis milling process. The chatter stability of the dynamic milling system is modeled at each segment location along the tool path. A multi-frequency solution for the chatter stability with a semi-discretization method is derived. Both the kinematic transformation and the variation of cutter-workpiece engagement geometry are considered in the chatter mechanism analysis. In addition, the workpiece structure is simplified as the cantilever beam for the modal analysis. The dynamic coupling effect of the workpiece and the tool is also discussed in Section 4.3. Moreover, chatter is avoided by parameter optimization based on stability prediction with the constraints of spindle speed and spindle acceleration. Finally, the finish machining of an S-curve part is carried out as the experimental cutting test. The comparisons of simulation and experimental results show that the proposed optimization algorithm can effectively stabilize processing and improve surface quality.