Relative Varying Dynamics Based Whole Cutting Process Optimization for Thin-walled Parts

Abstract

Thin-walled parts are typically difficult-to-cut components due to the complex dynamics in cutting process. The dynamics is variant for part during machining, but invariant for machine tool. The variation of the relative dynamics results in the difference of cutting stage division and cutting parameter selection. This paper develops a novel method for whole cutting process optimization based on the relative varying dynamic characteristic of machining system. A new strategy to distinguish cutting stages depending on the dominated dynamics during machining process is proposed, and a thickness-dependent model to predict the dynamics of part is developed. Optimal cutting parameters change with stages, which can be divided by the critical thickness of part. Based on the dynamics comparison between machine tool and thickness-varying part, the critical thicknesses are predicted by an iterative algorithm. The proposed method is validated by the machining of three benchmarks. Good agreements have been obtained between prediction and experimental results in terms of stages identification, meanwhile, the optimized parameters perform well during the whole cutting process.

1 Introduction

Thin-walled parts are widely used in airplane for many advantages such as reducing weight, improving space utilization and so on. The thickness of those parts is at least six times lower than the two other relevant directions, thus being flexible and easy to bend [1]. Moreover, large amount of materials (more than 90% of the blank) required to be removed during machining. There is an urgent need to improve efficiency. Those parts are mainly manufactured by high speed milling, where problems can arise related to chatter in the process [2]. The chatter is avoided by predicting stability lobe diagram either in frequency [3, 4] or discrete-time domain [5, 6] based on the structural dynamics of part and machine tool. During machining, the dynamics is variant for part, but invariant for machine tool. Therefore, the relative dynamics varies in cutting process, changing the conditions of stability.

The dynamics of part and machine tool can be obtained by many methods, including experimental modal analysis (EMA), finite element method (FEM), and so on. EMA is a commonly way to measure the dynamics. However, part is physically machined and the process is interrupted for measurements at discrete stations along the toolpath [7, 8], which is prohibitive in production. Meanwhile, the dynamics of machine tool changes with tool and holder, and it is time consuming to be tested by EMA. Therefore, many researchers investigated the methods to predict the dynamics of machine tool and part. Receptance coupling substructure analysis (RCSA) developed by Schmitz et al. [9,10,11,12] efficiently predicts the dynamics of the machine tool with different holders and tools. Zhang et al. [13] developed a method to predict the FRFs of tool point with arbitrary spindle orientations based on the frequency response function (FRF) measured in three orthogonal postures of the spindle. Besides, Budak et al. [14, 15] also made contribution to improve the RCSA method itself and its accuracy. For the dynamics prediction of part, Meshreki et al. [16] proposed a model based on representing the change of thickness with two-directional multispan plate. Fischer et al. [17] presented a flexible multibody system model for the inside turning of thin-walled cylinders. Ahmadi [18] proposed finite strip modeling for the dynamics prediction of thin-walled parts with complex geometries. Karimi et al. [19] analyzed the dynamics of rectangular plate subjected to a mass moving with variable velocity on a predefined path or an arbitrary one. Wang et al. [20] analyzed the influence factors on natural frequencies of composite materials. In order to improve computational efficiency, Cunedioğlu et al. [21] reduced the order of the FEM by implementing the frequency domain identification methods. Tuysuz and Altintas [22] proposed a frequency-domain model to predict the dynamics of in-process workpiece using reduced order substructuring method, which provides ~ 20 times faster FRF prediction than FEM. Then Tuysuz et al. [23] developed a time-domain model, and the new model is ~ 4 times more computationally efficient than the previous one.

The associated machining optimization based on dynamics have been investigated by many scholars. Altintas et al. [24] analytically predicted the stability lobes in milling based on the dynamics of machine tool. Davies et al. [25] proposed a stability theory for highly interrupted machining, which is always employed in high speed milling. Seguy et al. [26] developed an explicit numerical model to examine the relationship between chatter instability and surface roughness evolution. Kersting et al. [27] predicted regenerative vibrations during the five-axis milling process. Zhou et al. [28] presented an analytical chatter prediction model for bull-nose end milling of aero-engine casings. Liu et al. [29] proposed a prediction method for the stability of free-form surface milling. Shi et al. [30] predicted the thin-walled component milling stability considering material removing process. Bolsunovskly et al. [31] developed a parameters optimization method based on finite element model of part. Yi et al. [32] studied the deformation law and mechanism for milling micro thin wall with mixed boundaries, and obtained the corresponding optimal radial depth of cut and feed per tooth. Ringgaard et al. [33] maximized the material removal rate in milling of thin-walled parts without violating forced vibration and chatter stability. Gu et al. [34] presented three degrees of freedom dynamic model applied to tool chatter for thin-walled structures in milling. Jiang et al. [35] applied reliability analysis of a dynamic structural system to predict chatter of side milling system for machining blisk. Chen et al. [36] obtained force-deformation coupling relationship and time-based deformation matrix of thin-walled milling operation. Sanz-Calle et al. [37] studied the influence of radial engagement and milling direction on stability. For thin-walled part, Yao et al. [38] proposed a position-varying surface roughness prediction method. Ahmed et al. [39] developed a model to determine the part’s feasible location for the suitable setup parameters. Guo et al. [40] investigated the effects of feed rate on surface integrity in ultrasonically-assisted vertical milling. Limited researches have been done in the relative varying dynamics based machining. Meshreki et al. [41] mentioned the variation of dominant dynamics from roughing to finishing. Bravo et al. [2] developed a three-dimensional lobe diagram based on the dynamics variation of part to cover the intermediate stages in machining. Tuysuz and Altintas [22] depicted the invariant dynamics of tool and the varying dynamics of part at different stages.

Previous studies mainly focus on partial cutting process. However, the machining efficiency is evaluated for whole cutting process from blank to part, and optimization only for one stage of machining may trap in local optimum. Literatures have not reported the relative varying dynamics based whole cutting process optimization, which is of significance to production.

The paper proposes a novel method of whole cutting process optimization based on the relative varying dynamics of machining system. The strategy of dominated dynamics based cutting stage division and the thickness-dependent dynamics model of part are developed in Section 2. Section 3 presents the multi-variable function of machining efficiency of whole cutting process, and proposes the critical thickness solution method to distinguish stages. The experimental design, results are discussed in Section 4. Section 5 concludes the paper.

2 Relative Varying Dynamics During Thin-walled Part Machining

2.1 Dominated Dynamics Based Cutting Stage Division

Machining system composes of machine tool and part, and its dynamics is influenced by two sub-systems together.

$$[{\Delta }^{m} ] + [{\Delta }^{p} ] = [{\Delta }],$$

(1)

where [Δ^m], [Δ^p], and [Δ] represent the dynamics of machine tool, part, and machining system respectively. In three-axis milling with unchangeable tool and its overhang length, [Δ^p] changes with the material removal in cutting process, while [Δ^m] keeps constant.

The whole cutting process can be divided into three stages based on the relative dynamics between [Δp] and [Δ^m]. At the beginning (stage 1), as shown in Figure 1(a), the part is thick and can be regarded as rigid. With the material removal, the thickness and stiffness of part decrease, and the dynamics of two subsystems affects the machining process together (stage 2) as depicted in Figure 1(b). At the last stage (stage 3), the dynamics is dominated by part as illustrated in Figure 1(c). The relative dynamic characteristic varies with cutting process for thin-walled parts.

Figure 1

Relative varying dynamics in whole cutting process: (a) Dynamics dominated by machine tool, (b) Dynamics dominated by machine tool and part, (c) Dynamics dominated by part

In whole cutting process, the dynamics of machining system is rewritten as a piecewise function based on the thickness of part, expressed as

$$[{\Delta }] = \left\{ {\begin{array}{*{20}l} {[{\Delta }_{{}}^{m} ],} \hfill \\ {[{\Delta }_{{}}^{m} ] + [{\Delta }^{p} (t)],} \hfill \\ {[{\Delta }^{p} (t)],} \hfill \\ \end{array} \begin{array}{*{20}c} {} & {\begin{array}{*{20}l} {t > t_{1} ,} \hfill \\ {t_{1} \ge t \ge t_{2} ,} \hfill \\ {t_{2} > t,} \hfill \\ \end{array} } \\ \end{array} } \right.$$

(2)

where t represents the thickness of part; t₁ and t₂ are the critical thicknesses to divide stages; [Δ^m] can be obtained by the impact testing or RCSA method. However, [Δ^p] changes with the thickness.

2.2 Modelling of Thickness-dependent Dynamics of Thin-walled Parts

The thickness of part decreases with the material removal. When the dynamics of part influences the machining process, the aspect ratio of the length and height to the thickness is large enough so that the part can be modelled by using Kirchhoff’s thin plate theory, in which stress changes in the thickness direction is ignored. The thickness-dependent dynamics model is based on the assumption that the wall of part is reduced from both side simultaneously to remain the neutral plane unchanged during machining. The error of the assumption on the dynamics prediction is negligible [18], and four nodes quadrilateral element is used to mesh the neutral plane.

Based on the vibration of part and the Kirchhoff’s thin plate theory, the degree-of-freedom (DOF) of a node in the element can be reduced from 6 to 3, and the shape function of an element can be expressed as

$$\begin{array}{*{20}c} {{\varvec{N}} = \left[ {N_{{n_{1} }} } \right.} & {N_{{xn_{1} }} } & {N_{{yn_{1} }} } & {} & {} & {} \\ {} & {N_{{n_{2} }} } & {N_{{xn_{2} }} } & {N_{{yn_{2} }} } & {} & {} \\ {} & {} & {N_{{n_{3} }} } & {N_{{xn_{3} }} } & {N_{{yn_{3} }} } & {} \\ {} & {} & {} & {N_{{n_{4} }} } & {N_{{xn_{4} }} } & {\left. {N_{{yn_{4} }} } \right]} \\ \end{array} ,$$

(3)

$$\begin{gathered} \left\{ {\begin{array}{*{20}c} {N_{i} = \frac{1}{8}\left( {1 + \frac{x}{{x_{i} }}} \right)\left( {1 + \frac{y}{{y_{i} }}} \right)\left[ {2 + \frac{x}{{x_{i} }}\left( {1 - \frac{x}{{x_{i} }}} \right) + \frac{y}{{y_{i} }}\left( {1 - \frac{y}{{y_{i} }}} \right)} \right],} \\ {N_{xi} = - \frac{1}{8}y_{r} \left( {1 + \frac{x}{{x_{i} }}} \right)\left( {1 + \frac{y}{{y_{i} }}} \right)^{2} \left( {1 - \frac{y}{{y_{i} }}} \right),} \\ {N_{yi} = - \frac{1}{8}x_{r} \left( {1 + \frac{x}{{x_{i} }}} \right)^{2} \left( {1 + \frac{y}{{y_{i} }}} \right)\left( {1 - \frac{x}{{x_{i} }}} \right),} \\ \end{array} } \right. \hfill \\ \begin{array}{*{20}c} {(i = n_{1} ,n_{2} ,n_{3} ,n_{4} ),} & {} \\ \end{array} \hfill \\ \end{gathered}$$

(4)

where x_i and y_i (i=n₁, n₂, n₃, n₄) are the coordinates of node in the element coordinate system. B is the independent variable of the stiffness matrix, and is derived from Eqs. (3) and (4):

$${\varvec{B}} = \left[ {\begin{array}{*{20}c} {{\varvec{B}}_{{n_{1} }} } & {{\varvec{B}}_{{n_{2} }} } & {{\varvec{B}}_{{n_{3} }} } & {{\varvec{B}}_{{n_{4} }} } \\ \end{array} } \right],$$

(5)

$$\begin{array}{*{20}c} {} & \begin{gathered} {\varvec{B}}_{r} = - \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} N_{r} }}{{\partial x^{2} }}} & {\frac{{\partial^{2} N_{xr} }}{{\partial x^{2} }}} & {\frac{{\partial^{2} N_{yr} }}{{\partial x^{2} }}} \\ {\frac{{\partial^{2} N_{r} }}{{\partial y^{2} }}} & {\frac{{\partial^{2} N_{xr} }}{{\partial y^{2} }}} & {\frac{{\partial^{2} N_{yr} }}{{\partial y^{2} }}} \\ {2\frac{{\partial^{2} N_{r} }}{\partial xy}} & {2\frac{{\partial^{2} N_{xr} }}{\partial xy}} & {2\frac{{\partial^{2} N_{yr} }}{\partial xy}} \\ \end{array} } \right], \hfill \\ (r = n_{1} ,n_{2} ,n_{3} ,n_{4} ). \hfill \\ \end{gathered} \\ \end{array}$$

(6)

By using the virtual work principle, the stiffness matrix K^e and mass matrix M^e of element are expressed as the function of thickness.

$${\varvec{K}}^{e} (t) = \int {_{{V^{e} }} t^{3} {\varvec{B}}^\text{T} {\varvec{DB}}{\text{d}}V} ,$$

(7)

where D is the modified bending modulus, and is calculated as

$${\varvec{D}} = \frac{E}{{12(1 - \upsilon^{2} )}}\left[ {\begin{array}{*{20}c} 1 & \upsilon & 0 \\ \upsilon & 1 & 0 \\ 0 & 0 & {{{(1 - \upsilon )} \mathord{\left/ {\vphantom {{(1 - \upsilon )} 2}} \right. \kern-\nulldelimiterspace} 2}} \\ \end{array} } \right],$$

(8)

where E is the elasticity modulus, and υ is the Poisson ratio of materials.

$${\varvec{M}}^{e} (t) = e_{w} e_{l} \rho t\int {_{{S^{e} }} {\varvec{N}}^{{\text{T}}} {\varvec{N}}{\text{d}S}} ,$$

(9)

where e_w and e_l are the width and length of element; ρ is the density of material.

The damping matrix can be represented in terms of mass and stiffness matrices. For highly computational efficiency, Rayleigh damping is chosen to establish the damping matrix, which can be formulated as

$${\varvec{C}}^{e} (t) = \alpha {\varvec{M}}^{e} (t) + \beta {\varvec{K}}^{e} (t),$$

(10)

where α and β are damping coefficients identified from experiments.

To develop the matrices of part, the stiffness matrix, mass matrix and damping matrix of elements need to be assembled in nodes by using matrix displacement method. The dynamics model of the thickness-dependent dynamics of thin-walled part is

$${\varvec{M}}(t)\ddot{x} + {\varvec{C}}(t)\dot{x} + {\varvec{K}}(t){\varvec{x}} = {\varvec{F}}.$$

(11)

[Δ^p(t)] representing the direct FRFs of the cutting point is solved from the dynamics model, as expressed by

$$[\Delta^{p} (t)] = \sum\limits_{i = 1}^{n} {\frac{{{{\varvec{\uppsi}}}_{iq} (t){{\varvec{\uppsi}}}_{iq}^{\rm T} (t)}}{{(\omega_{i}^{2} (t) - \omega^{2} ) + 2j\zeta_{i} (t)\omega_{i} (t)\omega }}} ,$$

(12)

where q represents the DOF of lateral deflection in the weakest stiffness point. Ψ_iq and ω_i are the eigenvector and eigenvalue of the dynamics model respectively, ζ_i is the modal damping ratio, and n is the number of natural modes considered in synthesizing the FRF.

3 Whole Cutting Process Optimization

3.1 Material Removal Rates of Three Cutting Stages

Cutting conditions including the blank of part, the torque and power capabilities of machine tool, and the dynamics of machining system, determine the selection of cutting parameters. Boundary lines represented those conditions form the parameters domain as shown in Figure 2. The dynamics of machining system varies with stages, leading to the variation of the parameters domain.

Figure 2

Varying domain of appropriate parameters in whole cutting process

The mean material removal rate (MRR) is always used to evaluate the machining efficiency, given by

$$MRR(a_{e} ,a_{p} ,f_{t} ,N_{t} ,\Omega ) = a_{e} a_{p} f_{t} N_{t} \Omega ,$$

(13)

where a_e and a_p are the radial and axial depth of cut. f_t, N_t, and Ω represent the feed per tooth, the number of tooth, and the spindle speed, respectively. The combination of a_p and Ω determines the stability of machining process, which can be expressed as

$$a_{p} = f_{1} (k,[{\Delta }],a_{e} ,N_{t}^{*} ),$$

(14)

$$\Omega = f_{2} ([{\Delta }],a_{p} ),$$

(15)

where k and N_t^* represent the cutting force coefficient and the average number of teeth in the cut, respectively. [Δ] is expressed as a piecewise function based on the thickness of part. Substituting Eq. (2) into Eq. (13) yields MRR, which can be formulated as

$$MRR([{\Delta }]) = \left\{ {\begin{array}{*{20}l} {MRR_{1} ([{\Delta }_{{}}^{m} ]),} \hfill \\ {MRR_{2} ([{\Delta }_{{}}^{m} ] + [{\Delta }^{p} (t)]),} \hfill \\ {MRR_{3} ([{\Delta }^{p} (t)]),} \hfill \\ \end{array} \begin{array}{*{20}c} {} & {\begin{array}{*{20}l} {t > t_{1} ,} \hfill \\ {t_{1} \ge t \ge t_{2} ,} \hfill \\ {t_{2} > t,} \hfill \\ \end{array} } \\ \end{array} } \right.$$

(16)

where the subscripts (1, 2, 3) of MRR represent the cutting stages.

By dividing overall machining into 20 material removal steps with uniform volume, the dynamic stiffness of the part and machine tool are compared as shown in Figure 3. The part’s stiffness decreases due to the material removal, but machine tool’s stiffness keeps constant during the whole cutting process. The critical thicknesses t₁ and t₂, which are determined based on the stability curve comparison between the machine tool and the part with iterated thickness, divide the whole cutting process into three stages.

Figure 3

Stage division by critical thickness

The total volume of the removed material can be expressed as

$$v_{1} (t_{in} ,t_{1} ) + v_{2} (t_{1} ,t_{2} ) + v_{3} (t_{2} ,t_{fi} ) = v,$$

(17)

where t_in and t_fi are the thicknesses of the blank and part, and v is the volume of the material removal, which all can be obtained from the computer aided design (CAD) models. v₁, v₂ and v₃ are the removed volume of each stage, respectively.

The nominal total machining time (T_ma) is the sum of stage’s, and calculated as

$$\frac{{v_{1} (t_{in} ,t_{1} )}}{{MRR_{1} ([{\Delta }^{m} ])}} + \frac{{v_{2} (t_{1} ,t_{2} )}}{{MRR_{2} ([{\Delta }^{m} ] + [{\Delta }^{P} (t)])}} + \frac{{v_{3} (t_{2} ,t_{fi} )}}{{MRR_{3} ([{\Delta }^{P} (t)])}} = T_{ma} .$$

(18)

From Eq. (18), it is easy to conclude that the critical thickness t₁ and t₂ are the key parameters to determine the machining efficiency.

3.2 Critical Thickness Solution Method

The overall solution procedure includes identification module and computational modules as shown in Figure 4. In the identification module, the dimensions of the blank, and the elasticity modulus (E), density (ρ), and Poisson’s ratio (υ) of the material are recorded. The FRFs of the machine tool (tool point) are tested. The dimensions are used to update the height (h), length (l), and thickness (t) of the chosen plate. E, ρ, and υ are used to calculate the dynamics of part.

Figure 4

Flowchart of the critical thickness solution method

The computation module for t₁ is divided into four steps. Step 1: stability curve of the machine tool is obtained by using its FRFs and stability lobe diagram. The maximum (Max_M) and minimum (Min_M) of the curve is recorded. Step 2: The FRF of the part is predicted by substituting the E, ρ, and υ, the l, h, and t into the dynamic model developed in Section 2.2. t is divided by a, which is used to update the thickness in each iteration. Step 3: stability curve of the part is extracted and minimum (Min_P) is recorded. Step 4: h, l, and t are updated until Max_M equals to Min_P. However, it is difficult to obtain this solution exactly. Therefore, the iteration continues until the ratio of the Max_M and Min_P are in [b, d]. The corresponding thickness is the critical thickness t₁. The thickness (t₁) and corresponding length (l’), height (h’) are recorded.

In the computational module of t₂, the initial updated parameters are h’, l’, and t₁, and the process is divided into three steps. Step 1: the FRFs of the part is calculated based on the initial parameters, and the thickness is updated by dividing a', which is smaller than a to reduce the updated rate. Step 2: stability curve of the part is calculated and maximum (Max_P) is recorded. Step 3: if the value of Min_M divided by Max_P is in [b, d], the update of thickness will be terminated and the critical thickness t₂ is obtained.

4 Experimental Verification

4.1 Dynamics Testing of Machine Tool

The testing of dynamics of machine tool is the first step to calculate the critical thicknesses of part. A solid carbide end mill with a diameter of 12 mm and a length of 60 mm was equipped with the machining center DMU50 to machine benchmarks, as shown in Figure 5. The impact hammer (model: PCB 086C03) was used to excite the tool point in the X and Y directions, and the vibration responses were recorded by a low weight accelerometer (model: PCB 352C23). The data was analyzed by the acquisition system (model: Crystal CoCo-80X). The final measured FRFs of tool point was the mean value of the results of 5 times repeated tests. Modal parameters of the machine tool are listed in Table 1.

Figure 5

Dynamics testing of machine tool

Table 1 Modal parameters of the machine tool

4.2 Division of Cutting Stages

The machining process of a thin-walled pocket with 1 mm thickness (t), 30 mm height (h) was used to verify the proposed method. The dimensions of blank were 190 mm length (l), 120 mm width (w) as shown in Figure 6, and the material is Al7050 with E = 71.7 GPa, υ = 0.33, ρ = 2700 kg/m³.

Figure 6

Blank and benchmark

Three stages described in Section 2.1 are divided based on the dynamics of part and machine tool. Stability lobe diagrams were calculated by using semi-discretization method [5]. The tangential and radial cutting force coefficients were 8.6×10⁸ N/m² and 2.47×108 N/m² respectively. Down milling was used in machining, and the radial depth of cut was 6 mm, 1.5 mm, and 0.75 mm for stages 1, 2, and 3 respectively. The dynamics of machine tool and the dimension and material parameters of part were substituted into the method developed in Section 3.2. a, a’, b, and d were set as 2, 1.1, 0.8 and 1.2 respectively. t₁ and t₂ were 5 mm and 2.5 mm by calculation. The corresponding dimensional changes in three stages are illustrated in Figure 7, and the stability curves of machine tool and part in critical thicknesses are shown in Figure 8.

Figure 7

Dimensional changes in three stages

Figure 8

Stability curves of machine tool and part in critical thicknesses

4.3 Selection of Cutting Parameters

FRFs comparisons between the machine tool and the part in two critical thicknesses are shown in Figure 9, which clearly shows that the dominant natural frequencies of the machining system change with the cutting process. At the beginning, the dynamics of the machining system is dominated by the machine tool. When the thickness is between 5 mm and 2.5 mm, the dynamics of the machine tool and part should be both taken into account. In the last stage, the dynamics of the machining process was mainly determined by the part.

Figure 9

FRFs comparison of machine tool and part in critical thicknesses

Appropriate cutting parameters vary during machining, which should be optimized in each stage based on the dominated dynamics and corresponding stability lobe diagram as illustrated in Figure 10. The optimized cutting parameters in whole cutting process are listed in Table 2.

Figure 10

Dominated dynamics and corresponding stability lobe diagrams of three stages: (a) Stage 1, (b) Stage 1, (c) Stage 1

Table 2 Optimized cutting parameters in whole cutting process

4.4 Experimental Results

Benchmark was machined by the whole cutting process optimization method, and the machining results after three stages are shown in Figure 11. It is clearly seen that the whole cutting process was stable since the cutting parameters are determined by the stability lobe diagrams.

Figure 11

Machining results of Benchmark after three stages

Other two benchmarks (benchmark_1 and benchmark_2) are designed to test the accuracy of critical thickness. At stage 1, the thicknesses of two walls are set to 4.9 mm and 5.1 mm, respectively. At stage 2, they are set to 2.4 mm and 2.6 mm, respectively, as shown in Figure 12. Both the two benchmarks were machined by the cutting parameters of the corresponding stages in Table 2.

Figure 12

Machining process of: (a) Benchmark_1, (b) Benchmark_2

The machining results are illustrated in Figure 13. Compared to the previous results in Figure 11, the cutting instability both occurs at t = 4.9 mm and t = 2.4 mm, which indicates the accuracy of the calculated critical thickness.

Figure 13

Machining results of benchmark_1 and benchmark_2

5 Conclusions

The paper presents a novel method to realize whole cutting process optimization for the thin-walled parts based on relative varying dynamic characteristics of machining system. A number of conclusions can be drawn based on the derivation and validation of the proposed method.

• (1) For the thin-walled parts, the whole cutting process is divided into three stages: dynamics dominated by machine tool, both machine tool and part, and part respectively to match the change of stability caused by the relative varying dynamics of machining system.

• (2) The cutting parameters are optimized in each stage based on the analysis of stability to make full use of the capabilities of the machining system.

• (3) The critical thickness t₁ and t₂ are the key parameters to divide cutting stage and determine the machining efficiency. By using the proposed thickness-dependent dynamics model of part and iterative algorithm, the appropriate critical thicknesses are well predicted.

• (4) The method can be used in the whole cutting process optimization of various machining system without going through trial and error based cutting tests.