Multi-scale prediction of the geometrical deviations of the surface finished by five-axis ball-end milling

Abstract

Surface quality and accuracy are the main factors which affect the performance and life cycle of the products. Due to the complexity of the machining process, it is difficult to evaluate the machined surface real time. Simulation of the machining process became the main method to predict and control the quality of the machined surface. This article developed a multi-scale simulation system to predict the overall geometrical features of the milled surface. The effects of locating errors, geometrical errors of the machine tool and tool deflections on the quality of the machined surface are included in the proposed model. Also, different strategies are employed to evaluate the macro-scale and micro-scale geometrical deviations of the machined surface to balance the time cost and accuracy. In comparison with the traditional method, both the form deviations and roughness feature of the machined surface can be predicted. Since the static and dynamic properties of the machining system were considered, both the stable and unstable cutting conditions can be analyzed by using the proposed method. At the end of this article, case studies are carried out to validate the proposed method. The effects of the locating errors, geometrical errors of the machine tool and cutting parameters on the quality of the machined surface are analyzed. The significance of their influences on the quality of the machined surface was investigated.

Introduction

As the main concerns in manufacturing process, surface quality and accuracy are the main factors which affect the performance and life cycle of products. Due to the complexity of five-axis milling process, it is difficult to evaluate the machined surface real time. Simulation of the milling process became the main method to predict and control the quality of the machined surface. At the process-planning stage, the simulation results could be used to help the engineers to select proper process parameters, that is, depth of cut, cutting speed and feed speed to improve the quality of the machined surface.

Cutting force is the most fundamental, and in many cases, the most significant parameter in manufacturing process. In early studies, the milling force coefficients, which relate the forces to chip loads were developed from the experimental milling force tests. Among them, Martelotti ¹ modeled the complex geometry and relative part-tool motion in milling; Koenigsberger and Sabberwal ² developed equations for milling forces by using mechanistic modeling. Those mechanistic approaches have been applied to predict the associated machine component or surface geometrical errors.^3,4 Different with the traditional methods, an alternative or “unified mechanics of cutting” approach was proposed by Armarego and Whitfield ⁵ in 1985. After that, this approach has been developed by Budak and Altintas et al. in many researches.^6–10 In recent years, Ozturk and colleagues^11–13 and Altintas and Lee^14,15 and Altintas ¹⁶ analyzed the dynamics and stability of five-axis ball-end milling based on the “unified mechanics of cutting” approach.

Simulations of the milling process constitute an active research topic in the manufacturing community. Relevant published research works can be summarized as follows. Gao et al. ¹⁷ proposed a mesh-independent direct computing method for numerical simulation of machined surface topography and roughness in milling process. Campomanes and Altintas ¹⁸ proposed an improved milling time domain model to simulate vibratory cutting conditions at very small radial widths of cut. With this model, the cutting forces, surface finish and chatter stability can be predicted. Bouzakis et al. ¹⁹ developed a computer-supported milling simulation algorithm “Ballmill” to predict the roughness of the finished workpiece. Imani and colleagues^20,21 used solid modeling techniques and Boolean operations to deal with the geometrical simulation of the ball-end milling operations. Mizugaki et al. ²² presented a theoretical estimation method for the machined surface profile. In this method, the cusp heights at any points of the workpiece were derived from the geometrical relationship between the cutting edge movement and the normal line at the point. Zhang et al. ²³ developed a new and general iterative algorithm for the numerical simulation of the machined surface topography in multi-axis ball-end milling based on the tool machining paths and the trajectory equation of the cutting edge relative to the workpiece. Liu et al. ²⁴ developed a comprehensive simulation system based on a Z-map model for predicting surface topographic features and roughness formed in the finish milling process. Jung et al.^25,26 proposed the ridge method for the prediction of the machined surface roughness in the ball-end milling process. Budak^27,28 presented analytical models to predict the milling force, part and tool deflection, form error and stability of the cutting process. In the works of Arizmendi et al.,^29–31 the effects of tool setting error, tool parallel axis offset, and tool vibration on the machined surface were considered in the modeling and simulation of surface topography in the milling process.

According to the geometrical product specifications (GPS), surface deviations are made up of form deviation, waviness and roughness. Where form deviation, waviness and roughness belong to the macro-scale, meso-scale and micro-scale geometrical deviations, respectively. Due to different causes of them, their effects on the performance of the products vary too. Simulation of the amplitude field of the machined surface, identification of the parameters and properties of the surface topography are of great importance to improve the surface quality.

In literature review, although there have been numerous works in process simulation, most of them cannot provide a multi-scale prediction of the amplitude field of the machined surface. This article developed a multi-scale simulation system to predict both the macro-scale and micro-scale geometrical deviations, that is, the form deviation and roughness of the machined surface. (Due to the fact that the GPS has not clearly defined the cut-off wavelengths of the waviness, it can be predicted by using the prediction strategy for the form deviation or roughness depending on the cut-off wavelengths set by the engineers.) The system is developed based on a multi-scale model of the workpiece. And the effects of locating errors, geometrical errors of the machine tool and tool deflections on the surface quality are included. Once the topography of the machined surface is obtained, the causes of those defects can be analyzed, the undesired results can be overcome and the quality and productivity can be improved by adjusting the related process parameters.

The organization of this article is as follows. In section “Multi-scale modeling of the workpiece,” authors’ former works about the multi-scale modeling method of the workpiece are reviewed. Then, the effects of locating errors and geometrical errors of the machine tool are analyzed in section “Modeling of the milling process.” The method for multi-scale predicting of the geometrical deviations of the machined surface is presented in section “Multi-scale prediction of the finished surface.” In section “Case studies,” case studies are carried out to validate the proposed method, and the effects of the locating errors, geometrical errors of the machine tool and cutting parameters on the quality of the machined surface are analyzed. The conclusions are summarized in section “Conclusion.”

Multi-scale modeling of the workpiece

A proper geometrical model of the workpiece is the foundation for multi-scale modeling of the milling process. A multi-scale geometrical model which was presented in authors’ former work is adopted to describe the rough workpiece. This method is based on the discrete modal decomposition (DMD) of the surface defects. ³² To balance the time cost and accuracy, different scales of deviations, that is, location/orientation deviation, form deviation, waviness and roughness, are displayed in different scales of areas.

The rough workpiece deviations are decomposed into different scales of deviations. Each scale of deviations is expressed as the products of the normalized deviations and deviation factors. The normalized deviations are built by using the DMD method. The deviation factors are calculated based on the analyzing of the tolerance values and the tolerancing principle of the workpiece. The final rough surface texture is formed by the superposition of each scale of deviations. Details of this method can be found in authors’ former work. ³³

Modeling of the milling process

Assumptions

There are many error sources which affect the accuracy and quality of the machined surface, such as geometrical errors of the machine tool, cutting force–induced errors, locating errors, thermal errors, spindle motion errors, vibrations and controller errors. It is very difficult to develop a comprehensive model which can include the relationships between all error sources. To simplify the modeling process, the following assumptions are introduced:

• The workpiece, machine tool and fixture are rigid, and their deformations are ignored. Usually, the slender cutting tool which has weak stiffness is the vulnerable part in a machining system. Compared with the cutting tool, the machine tool and fixture are assumed to be composed of rigid components. Their deformations are ignored in the modeling process. The workpiece is assumed to be rigid. Therefore, the proposed model is not valid for compliant workpiece whose deformations cannot be ignored under the acting of cutting force.

• Deviations of the workpiece and fixture and the geometrical errors of the machine tool are small. Under the assumptions of small errors, the locating errors of the workpiece and the geometrical errors of the machine tool can be represented by homogeneous transformation matrices, where the trigonometric functions of small geometrical errors can be simplified as constant or linear functions. The modeling process can be greatly simplified.

• The thermal errors are ignored. In a machining process, the variation in the ambient temperature and the heat generated in metal cutting and machine tool running will cause thermal deformations of the workpiece and additional geometrical errors of the machine tool. The existences of thermal errors increase the complexity of the machining process. In this article, it is assumed that the ambient temperature is kept constant; coolant liquid which can take away most of the heat generated in metal cutting is used in the operations, and the geometrical errors of the machine tool are measured after the machine tool was warmed up. Based on those assumptions, the thermal errors are ignored in the modeling process.

• Backlash errors and control errors of the movements of the axes of the machine tool are ignored.

• Uses pins or locating planes as fixture locators. In a machining process, the workpiece is initially positioned by a fixture. In this article, fixture with 3-2-1 locating scheme which is equivalent to all kinematics constraints on the workpiece produced by fixture is supposed to be used. ³⁴ The fixture locaters are simplified as pins and/or planes. One pin constraints 1 degree of freedom of the workpiece, and one plane constraints 3 degrees of freedom of the workpiece.

Besides these assumptions, only the effects of locating errors, geometrical errors of the machine tool and tool deflections are studied in this article. Although other types of errors are not included, interfaces to them are included in the developed model, which will be presented in the following parts.

Cutter geometry and coordinate system definitions

In the modeling of five-axis ball-end milling, familiarity with the tool geometry and coordinate systems is needed. They will be explained briefly in this section.

As shown in Figure 1, a tool coordinate system (TCS) is defined at the ball center. The Z_T-axis is along the tool axis direction. The radius of the ball is R₀. For a local point q at elevation z on a cutting edge, the local radius, axial immersion angle and radial lag angle are represented as R(z), K(z) and Ψ(z), respectively. φ_p is the pitch angle of the tool. The immersion angle of the jth flute φ_j(z) is measured from the positive direction of Y_T, and φ_j(z) = φ + (j − 1)φ_p − Ψ(z). The uncut chip thickness h at a point on the cutting edge can be approximated by the scalar product of the displacement of the ball-end mill d and the unit outward surface normal vector u of the point

$$h=\mathit{u}·\mathit{d}$$

(1)

Figure 1.

Geometry of the ball-end mill.

The displacement of the ball-end mill is defined as the difference between the current coordinates of the ball center (x(t), y(t), z(t)) and the coordinates one tooth period before (x(t − Δt), y(t − Δt), z(t − Δt)) in the TCS

$$\mathit{d}=\left[\begin{array}{c}\hfill x(t)-x(t-\Delta t)\hfill \\ \multicolumn{1}{c}{y(t)-y(t-\Delta t)}\\ \multicolumn{1}{c}{z(t)-z(t-\Delta t)}\end{array}\right]$$

(2)

The unit outward surface normal vector u is calculated in the TCS as

$$\mathit{u}=\{\begin{array}{c}\hfill \frac{1}{R_0}·[R(z)·\sin ({\phi }_j(z)),R(z)·\cos ({\phi }_j(z)),z],z\le 0\hfill \\ \multicolumn{1}{c}{[\sin ({\phi }_j(z)),\cos ({\phi }_j(z)),0],z>0}\end{array}$$

(3)

In addition to TCS, two additional coordinate systems, namely, machine coordinate system (MCS) and FCN, are introduced to define the position and orientation of the cutting tool. As shown in Figure 2, MCS is fixed on the machine tool. In FCN, F represents the feed direction, N is the normal direction of the machined surface and C is the cross-feed direction. Cross-feed direction is defined as follows: if uncut material is in the positive C-axis with respect to the cutting tool, the cross-feed direction is positive; if uncut material is in the negative C-axis with respect to the cutting tool, the cross-feed direction is negative.

Figure 2.

Coordinate frames of five-axis ball-end mill: (a) configuration of the machine tool, (b) FCN coordinate system and (c) lead and tilt angles.

A transformation matrix T_F,T is used to transform the coordinates, forces and displacements from TCS to FCN

$${\mathit{V}}_{\mathit{F},\mathit{T}}={\mathbf{T}}_{\mathbf{F},\mathbf{T}}·{\mathit{V}}_{\mathit{T},\mathit{T}}$$

(4)

T_F,T is defined in equation (5), where l_a and t_a are the lead and tilt angles, respectively

$${\mathbf{T}}_{\mathbf{F},\mathbf{T}}=\left[\begin{array}{ccc}\hfill 1\hfill & 0& 0\\ \multicolumn{1}{c}{0}& \cos t_a& -\sin t_a\\ \multicolumn{1}{c}{0}& \sin t_a& \cos t_a\end{array}\right]\left[\begin{array}{ccc}\hfill \cos l_a\hfill & 0& \sin l_a\\ \multicolumn{1}{c}{0}& 1& 0\\ \multicolumn{1}{c}{-\sin l_a}& 0& \cos l_a\end{array}\right]$$

(5)

The effects of locating errors

In a milling process, the workpiece is initially positioned by a fixture. Due to the deviations on the fixture and the datum of workpiece, the workpiece would deviate from its nominal position in the fixture. The existence of locating errors increases the dimension and location errors of the machined workpiece. In this article, the translation and rotation deviations of the workpiece caused by the locating errors are represented by a homogeneous transformation matrix T_W0,W as ³⁴

$${\mathbf{T}}_{\mathbf{W}0,\mathbf{W}}=\left[\begin{array}{cccc}\hfill 1\hfill & \Delta \gamma & -\Delta & \beta \Delta x\\ \multicolumn{1}{c}{-\Delta \gamma }& 1& \Delta \alpha & \Delta y\\ \multicolumn{1}{c}{\Delta \beta }& -\Delta \alpha & 1& \Delta z\\ \multicolumn{1}{c}{0}& 0& 0& 1\end{array}\right]$$

(6)

where (Δx, Δy, Δz) and (Δγ, Δβ, Δα) represent the translation and the rotation deviation respectively.

T_W0,W links the designed and actual positions of the workpiece in the fixture. To obtain the parameters in T_W0,W, the contact points between the fixture and the workpiece must be identified. As shown in Figure 3, when pins are used as fixture locaters, it is easy to obtain the contact points. For locating planes, an approach based on the convex hull theory is introduced here to identify the contact points. It can be stepped as follows:

• Determination of the difference face Λ. The difference face is defined as the difference between the fixture surface and datum surface: Λ = Λ _d  − Λ _f .

• Identification of the possible contact points. Construct the convex hull of Λ, the contact facet is the one which intersects the clamping force or the gravity. Its vertexes are the contact points.

Figure 3.

Locating errors of the workpiece: (a) fixture with pins and (b) fixture with locating planes.

For the ith contact point, the point on the datum is denoted as P_i , its normal vector as n_i and the deviations on the fixture and datum of workpiece as Δf_i and Δd_i, respectively. Then, the contact error can be written as Δc_i =Δf_i + Δd_i. The deviations of the pins and/or the locating planes Δf_i and deviations of the datum Δd_i can be obtained from measurement by using a dial gauge or Coordinate Measuring Machine (CMM). For a fixture with 3-2-1 locating scheme, the relationship between the contact errors and the locating errors can be written as

$$\left[\begin{array}{c}\hfill \Delta c_1\hfill \\ \multicolumn{1}{c}{\Delta c_2}\\ \multicolumn{1}{c}{\Delta c_3}\\ \multicolumn{1}{c}{\Delta c_4}\\ \multicolumn{1}{c}{\Delta c_5}\\ \multicolumn{1}{c}{\Delta c_6}\end{array}\right]=-\left[\begin{array}{cc}\hfill {\mathit{n}}_1^{\mathit{T}}\hfill & ({\mathit{P}}_1\times {\mathit{n}}_1{)}^T\\ \multicolumn{1}{c}{{\mathit{n}}_2^{\mathit{T}}}& ({\mathit{P}}_2\times {\mathit{n}}_2{)}^T\\ \multicolumn{1}{c}{{\mathit{n}}_3^{\mathit{T}}}& ({\mathit{P}}_3\times {\mathit{n}}_3{)}^T\\ \multicolumn{1}{c}{{\mathit{n}}_4^{\mathit{T}}}& ({\mathit{P}}_4\times {\mathit{n}}_{\mathit{4}}{)}^T\\ \multicolumn{1}{c}{{\mathit{n}}_{\mathit{5}}^{\mathit{T}}}& ({\mathit{P}}_{\mathit{5}}\times {\mathit{n}}_{\mathit{5}}{)}^T\\ \multicolumn{1}{c}{{\mathit{n}}_{\mathit{6}}^{\mathit{T}}}& ({\mathit{P}}_{\mathit{6}}\times {\mathit{n}}_4{)}^T\end{array}\right]\left[\begin{array}{c}\hfill \Delta x\hfill \\ \multicolumn{1}{c}{\Delta y}\\ \multicolumn{1}{c}{\Delta z}\\ \multicolumn{1}{c}{\Delta \alpha }\\ \multicolumn{1}{c}{\Delta \beta }\\ \multicolumn{1}{c}{\Delta \gamma }\end{array}\right]$$

(7)

From equation (7), the parameters in equation (6) can be obtained. To further investigate the effects of geometrical errors of machine tool and the tool deflections on the machined surface, the coordinates of the workpiece should be further transformed from workpiece coordinate system (WCS) to MCS

$${\mathit{V}}_{\mathit{M},\mathit{W}}={\mathbf{T}}_{\mathbf{M},\mathbf{W}0}·{\mathbf{T}}_{\mathbf{W}0,\mathbf{W}}·{\mathit{V}}_{\mathit{W},\mathit{W}}$$

(8)

T_M,W0 is determined by the designed position of the workpiece in the machine tool.

The effects of geometrical error of machine tool

A five-axis milling machine tool has 37 independent geometrical error components when the machine tool is considered as a set of rigid bodies. If a coordinate system is fixed to each of the five bodies, there can be 6 errors per body or 30 errors. The errors due to the non-orthogonality of the MCSs or squareness errors have seven independent components.

As shown in Figure 4, to determine the positions of the tool and workpiece in the machine tool, a group of additional coordinate systems are defined. All the X-reference axes are coinciding with the slide X. So the slide X does not have angular error or squareness error component. The X–Y plane of the machine tool is selected as the reference plane. So slide Y has one squareness error component γ_SY. Slide Z has two squareness error components β_SZ and α_SZ. Axis A is not exactly parallel with the slide Z. So it has two squareness error components β_SA and α_SA. Similarly, axis B has two squareness error components γ_SA and α_SA. The backlash errors and thermal deformations of the five axes are not considered in this article (Figure 5).

Figure 4.

The setup of the additional coordinate systems.

Figure 5.

The six geometrical error components of the translation and rotation axes: (a) component errors of Z-axis and (b) component errors of A-axis.

When the tool moves along the slides or rotate around the axes, small angular and linear errors will be added to the nominal translations and rotations. Under the assumption of rigid body behaviors, these errors are functions of the nominal movements only. With consideration of those error components, the coordinates of the tool can be transformed from TCS to MCS as

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{V}}_{\mathit{M},\mathit{T}}={\mathbf{T}}_{\mathbf{M},\mathbf{X}}·{\mathbf{T}}_{\mathbf{X},\mathbf{Y}}·{\mathbf{S}}_{\mathbf{Y}}·{\mathbf{T}}_{\mathbf{Y},\mathbf{Z}}·{\mathbf{S}}_{\mathbf{Z}}·{\mathbf{T}}_{\mathbf{Z},\mathbf{A}}·{\mathbf{S}}_{\mathbf{A}}·{\Omega }_A·{\mathbf{S}}_{\mathbf{B}}·{\mathbf{\Omega }}_{\mathbf{B}}·{\mathit{V}}_{\mathit{T},\mathit{T}}}\\ \multicolumn{1}{c}{;\mathrm{or}:{\mathit{V}}_{\mathit{M},\mathit{T}}={\mathbf{T}}_{\mathbf{M},\mathbf{T}}·{\mathit{V}}_{\mathit{T},\mathit{T}}}\end{array}$$

(9)

Details of the matrices in equation (9) are presented as follows

$${\mathbf{T}}_{\mathbf{M},\mathbf{X}}=\left[\begin{array}{cccc}\hfill 1\hfill & 0& 0& X_{M,X}\\ \multicolumn{1}{c}{0}& 1& 0& 0\\ \multicolumn{1}{c}{0}& 0& 1& 0\\ \multicolumn{1}{c}{0}& 0& 0& 1\end{array}\right]$$

(10a)

$${\mathbf{T}}_{\mathbf{I},\mathbf{J}}=\left[\begin{array}{cccc}\hfill 1\hfill & -\gamma (i)& \beta (i)& X_{I,J}+\Delta x(i)\\ \multicolumn{1}{c}{\gamma (i)}& 1& -\alpha (i)& Y_{I,J}+\Delta y(i)\\ \multicolumn{1}{c}{-\beta (i)}& \alpha (i)& 1& Z_{I,J}+\Delta z(i)\\ \multicolumn{1}{c}{0}& 0& 0& 1\end{array}\right]$$

(10b)

$${\mathbf{S}}_{\mathbf{K}}=\left[\begin{array}{cccc}\hfill 1\hfill & -{\gamma }_{\mathrm{SK}}& {\beta }_{\mathrm{SK}}& 0\\ \multicolumn{1}{c}{{\gamma }_{\mathrm{SK}}}& 1& -{\alpha }_{\mathrm{SK}}& 0\\ \multicolumn{1}{c}{-{\beta }_{\mathrm{SK}}}& {\alpha }_{\mathrm{SK}}& 1& 0\\ \multicolumn{1}{c}{0}& 0& 0& 1\end{array}\right]$$

(10c)

$${\mathbf{\Omega }}_{\mathbf{B}}=\left[\begin{array}{cccc}\cos \left({\theta }_B+\beta ({\theta }_B)\right)& -\gamma ({\theta }_B)& \sin \left({\theta }_B+\beta ({\theta }_B)\right)& \Delta x({\theta }_B)+X_{B,T}\\ \begin{array}{l}\gamma ({\theta }_B)\cos \left({\theta }_B+\beta ({\theta }_B)\right)+\\ \alpha ({\theta }_B)\sin \left({\theta }_B+\beta ({\theta }_B)\right)\end{array}& 1& \begin{array}{l}\gamma ({\theta }_B)\sin \left({\theta }_B+\beta ({\theta }_B)\right)-\\ \alpha ({\theta }_B)\cos \left({\theta }_B+\beta ({\theta }_B)\right)\end{array}& \Delta y({\theta }_B)+Y_{B,T}\\ -\sin \left({\theta }_B+\beta ({\theta }_B)\right)& \alpha ({\theta }_B)& \cos \left({\theta }_B+\beta ({\theta }_B)\right)& \Delta z({\theta }_B)+Z_{B,T}\\ 0& 0& 0& 1\end{array}\right]$$

(10d)

$${\mathbf{\Omega }}_{\mathbf{A}}=\left[\begin{array}{cccc}\cos \left({\theta }_A+\gamma ({\theta }_A)\right)& -\sin \left({\theta }_A+\gamma ({\theta }_A)\right)& \beta ({\theta }_A)& \Delta x({\theta }_A)+X_{A,B}\\ \sin \left({\theta }_A+\gamma ({\theta }_A)\right)& \cos \left({\theta }_A+\gamma ({\theta }_A)\right)& -\alpha ({\theta }_A)& \Delta y({\theta }_A)+Y_{A,B}\\ \begin{array}{l}\alpha ({\theta }_A)\sin \left({\theta }_A+\gamma ({\theta }_A)\right)-\\ \beta ({\theta }_A)\cos \left({\theta }_A+\gamma ({\theta }_A)\right)\end{array}& \begin{array}{l}\beta ({\theta }_A)\sin \left({\theta }_A+\gamma ({\theta }_A)\right)+\\ \alpha ({\theta }_A)\cos \left({\theta }_A+\gamma ({\theta }_A)\right)\end{array}& 1& \Delta z({\theta }_A)+Z_{A,B}\\ 0& 0& 0& 1\end{array}\right]$$

(10e)

In those equations (I, J) = (X, Y), (Y, Z), (Z, A); K = Y, Z, A, B. (X_I, _J , Y_I, _J , Z_I, _J ) represent the coordinates of the origin of J in I. (α_SK, β_SK, γ_SK) represent the squareness error components of K. The geometrical error components included in equation (10) can be obtained with the help of a laser interferometer and a double ball bar.^35,36 Consequently, the transformation matrix between TCS and WCS can be written as

$${\mathbf{T}}_{\mathbf{W},\mathbf{T}}=({\mathbf{T}}_{\mathbf{M},\mathbf{W}0}·{\mathbf{T}}_{\mathbf{W}0,\mathbf{W}}{)}^{-1}·{\mathbf{T}}_{\mathbf{M},\mathbf{T}}$$

(11)

In the milling process, the cutting edges would coincide with the points on the workpiece, that is, V_W , _T  =  V_W , _W . So, the points in cut with the cutting edge can be expressed as

$$\left[\begin{array}{c}\hfill X_{W,W}\hfill \\ \multicolumn{1}{c}{Y_{W,W}}\\ \multicolumn{1}{c}{Z_{W,W}}\end{array}\right]={\mathbf{T}}_{\mathbf{W},\mathbf{T}}·\left[\begin{array}{c}\hfill X_{T,T}\hfill \\ \multicolumn{1}{c}{Y_{T,T}}\\ \multicolumn{1}{c}{Z_{T,T}}\end{array}\right]$$

(12)

Besides the effects of locating errors and the geometrical errors of the machine tool, the static and dynamic deflections of the tool also affect the quality and accuracy of the machined surface. With consideration of the tool deflections, the machined surface can be finally expressed as

$${\mathit{V}}_{\mathit{W},\mathit{W}}={\mathbf{T}}_{\mathbf{W},\mathbf{T}}·({\mathit{V}}_{\mathit{T},\mathit{T}}+{\mathbf{\epsilon }}_{\mathit{T},\mathit{T}})$$

(13)

The effects of other error sources

As mentioned in section “Assumptions,” only the effects of locating errors, geometrical errors of the machine tool and tool deflections on the quality of the machined surface are included in the modeling of milling process. Although other types of errors are not considered in this article, interfaces to them are included. For example, the model of the geometrical errors of the machine tool (equations (9)–(11)) can be extended to include the thermal errors of the machine tool, the spindle errors and the setting errors of the tool; the form deviations of the tool can be added to ε _{T, T} in equation (13). To an extent, it is an open model to include the effects of other errors on the quality of the machined surface.

Multi-scale prediction of the finished surface

In section “Modeling of the milling process,” the effects of locating errors and geometrical errors of the machine tool have been modeled and analyzed. In this part, the prediction methods for the form deviations and roughness of the machined surface are developed, and the effects of the tool deflection are taken into account. To balance the time cost and accuracy of the simulation, two different strategies are employed to evaluate the geometrical deviations of the machined surface:

• Form deviations. Only the static parts of the tool deflection are considered in the model; the simulation area is the entire machined surface.

• Roughness. The effects of dynamic deflections of the tool are included in the model; the simulation area is limited to small part of the machined surface.

Modeling of cutting force

Reliable prediction of the cutting force is essential for determining the tool deflections. The cutting force model proposed by Lee and Altintas ⁶ is employed here. To calculate the cutting force, the cutter is divided into differential cutting elements as shown in Figure 6(a). The differential cutting forces in radial, tangential and axial directions are calculated by using the local cutting force coefficients (K_rc, K_tc, K_ac) and the edge force coefficients (K_re, K_te, K_ae) as

$$\{\begin{array}{c}\hfill {\mathrm{dF}}_t=K_{\mathrm{te}}·\mathrm{dS}+K_{\mathrm{tc}}·h·\mathrm{db}\hfill \\ \multicolumn{1}{c}{{\mathrm{dF}}_r=K_{\mathrm{re}}·\mathrm{dS}+K_{\mathrm{rc}}·h·\mathrm{db}}\\ \multicolumn{1}{c}{{\mathrm{dF}}_a=K_{\mathrm{ae}}·\mathrm{dS}+K_{\mathrm{ac}}·h·\mathrm{db}}\end{array}$$

(14)

where dS, h and db are the differential chip length, uncut chip thickness and chip length, respectively. The cutting force coefficients can be calculated as

$$\{\begin{array}{c}\hfill K_{\mathrm{tc}}=\frac{\tau }{\sin {\varphi }_n}·\frac{\cos ({\beta }_n-{\alpha }_n)+\tan i·\tan \eta ·\sin {\beta }_n}{\sqrt{{\cos }^2({\varphi }_n+{\beta }_n-{\alpha }_n)+{\tan }^2\eta ·{\sin }^2{\beta }_n}}\hfill \\ \multicolumn{1}{c}{K_{\mathrm{rc}}=\frac{\tau }{\sin {\varphi }_n·\cos i}·\frac{\sin ({\beta }_n-{\alpha }_n)}{\sqrt{{\cos }^2({\varphi }_n+{\beta }_n-{\alpha }_n)+{\tan }^2\eta ·{\sin }^2{\beta }_n}}}\\ \multicolumn{1}{c}{K_{\mathrm{ac}}=\frac{\tau }{\sin {\varphi }_n}·\frac{\cos ({\beta }_n-{\alpha }_n)·\tan i-\tan \eta ·\sin {\beta }_n}{\sqrt{{\cos }^2({\varphi }_n+{\beta }_n-{\alpha }_n)+{\tan }^2\eta ·{\sin }^2{\beta }_n}}}\end{array}$$

(15)

where τ is the shear stress, ${\varphi }_n$ is the normal shear angle, β_n is the normal friction angle, α_n is the normal rake angle, η is the chip flow angle and i is the local helix angle. All those factors and the edge force coefficients (K_re, K_te, K_ae) can be obtained from orthogonal cutting tests using oblique cutting model. ⁷ The calculated differential forces are transformed to TCS, and they are integrated over the engagement region to calculate the total cutting forces.

Figure 6.

Discretization of (a) the cutter and (b) the workpiece.

Ozturk and Budak ¹² introduced the engagement conditions for five-axis ball-end milling in their former work. However, those engagement conditions were developed in the ideal milling conditions. When the tool deflection is taken into account, the cutting edge may loss contact with the workpiece, or lead to the overcut of the workpiece. Then, those engagement conditions will not be valid.

In this article, a new method was developed to determine the engagement conditions. First, the common engagement conditions are as follows: chip thickness h is greater than 0 and the tool’s lowest point (cutter contact point) is lower than the top of the workpiece. In addition, a group of reference sections L_j (j = 1, …, n) are used to determine the engagement boundaries.

As shown in Figure 6(b), three matrices MT, MB, and MS are used to record the contours of the reference sections. The initial deviations of the top surface are stored in matrix MT. The other two matrices are used to store the contours left by the movements of the cutter.

At beginning, MB = MT, and MS stores the initial deviations of the surface in the cross-feed direction. The contours of the reference sections or matrices MB and MS are updated when the cutting edges are penetrating into the workpiece. Figure 7 illustrates the update process of those matrices. When the cutting tool is in cut with reference section L_j, the corresponding contour P_AP_BP_CP_D will be left on L_j. The C coordinates of the points on contour P_AP_BP_C will be stored in MS. The N coordinates of the points on contour P_BP_CP_D will be stored in MB. Based on those stored values, the engagement criteria are developed and listed in Table 1. Where N_o is the N coordinate of the ball center.

Figure 7.

Update of the machined surface.

Table 1.

Engagement criteria for the cutting tool.

Case	Cross-feed direction	N coordinate	Condition for engagement
1	Positive	No < N < MT(F, C)	C > MS(F, N)
2	Positive	N < No	N < MB(F, C)
3	Negative	No < N < MT(F, C)	C < MS(F, N)
4	Negative	N < No	N < MB(F, C)

In the simulation, the movements of the cutting tool and the machined surface are discretized. The cutting edges may not intersect the grid lines of the workpiece exactly or miss some points swept by the tool at certain positions. Therefore, interpolation method is adopted to solve those limitations. For each flutes, two arrays are employed to record their positions. As shown in Figure 8, one is used to record the last position $P_1^{\prime }{P}_2^{\prime }$ , and the other one is used to record the current position P₁ P₂. Since the coordinates of four corners are obtained, the coordinates of the points inside the quadrangle can be determined. And the missed points can be updated.

Figure 8.

Interpolation of the engagement regions.

Once the engagement regions are determined, the differential cutting force can be calculated by using equation (14). And they are transformed to TCS as

$$\left[\begin{array}{c}d{F}_X\\ d{F}_Y\\ d{F}_Z\end{array}\right]=\left[\begin{array}{ccc}-\sin K(z)\sin {\phi }_j(z)& -\cos {\phi }_j(z)& -\cos K(z)\sin {\phi }_j(z)\\ -\sin K(z)\cos {\phi }_j(z)& \sin {\phi }_j(z)& -\cos K(z)\cos {\phi }_j(z)\\ \cos K(z)& 0& -\sin K(z)\end{array}\right]\left[\begin{array}{c}d{F}_t\\ d{F}_r\\ d{F}_a\end{array}\right]$$

(16)

The calculated differential forces are integrated over the engagement region to calculate total cutting forces (F_X, F_Y, F_Z). The total cutting forces can be further transformed to WCS as

$$\left[\begin{array}{c}\hfill {\mathbf{F}}_{\mathit{WCS}}\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]=\left[\begin{array}{cc}\hfill {\mathbf{T}}_{\mathbf{W},\mathbf{T}}\hfill & 0\\ \multicolumn{1}{c}{0}& 1\end{array}\right]\left[\begin{array}{c}\hfill {\mathit{F}}_{\mathit{TCS}}\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]$$

(17)

Multi-scale prediction of the finished surface

As mentioned at the beginning of section “Multi-scale prediction of the finished surface,” to balance the time cost and accuracy of the simulation, two different strategies are employed to evaluate the form deviations and roughness of the machined surface. The details of those strategies are presented in this section.

Prediction of the form deviations

As mentioned before, form deviations reflect the macro-scale geometrical quality of a machined surface. Usually, it is maintained by using conservative cutting parameters such as low depth of cut or by compensation. Those approaches increase the time cost and decrease the productivity of the products. If the form deviations can be predicted before the machining, those shortcomings can be avoided. In the prediction of form deviations of the machined surface, only the static deflections of the tool are considered for simplicity; however, the deflections of the workpiece can also be determined with the help of finite element analysis.

A slender ball-end mill with a gauge length of L can be modeled as a cantilever beam with elastic supports as shown in Figure 9. The cutter contact point is defined as the ball-end mill’s lowest point in normal direction which is in cut with the workpiece. Denote the Z-coordinate of the cutter contact point in the TCS as z_k. Then, the differential tool deflection at the cutter contact point caused by the force applied at z_m in the TCS is given by the cantilever beam formulation as ³⁷

$${\delta }_i(z_k,z_m)=\{\begin{array}{l}\frac{d{F}_{i,m}{\nu }_m^2}{6EI}(3{\nu }_m-{\nu }_k)+\frac{1}{k_c},\mathrm{\hspace{1em}}0<{\nu }_k<{\nu }_m\\ \frac{d{F}_{i,m}{\nu }_m^2}{6EI}(3{\nu }_k-{\nu }_m)+\frac{1}{k_c},\mathrm{\hspace{1em}}{\nu }_m<{\nu }_k\end{array}$$

(18)

where i represents the deflection direction, E is the Young’s modulus, I is the area moment of inertia of the tool, ν_k = L − z_k, L is the distance from the ball center to the collet and k_c is the tool clamping stiffness in the collet. The ball-end mill is rigid along the tool axis direction; hence, the flexibilities of the tools in axis directions are neglected. The total tool deflection at the cutter contact point can be calculated as

$${\delta }_i(z_k)=\sum _{z_m}{\delta }_i(z_k,z_m)$$

(19)

When the deflections of the tool are obtained, ε _{T, T} in equation (13) can be evaluated as (δ_x(z_k), δ_y(z_k), 0)^T. Then, the form deviation can be evaluated by using the least square method.

Figure 9.

Structural model of the tool.

Prediction of the roughness

Besides the static deflections, the ball-end mill can deviate from its desired positions due to dynamic effects. The dynamic deflections of the tool can be caused by various sources such as slide speed variations, forced and self-excited vibrations. The dynamic effects of the tool can lead to periodic errors such as waviness and roughness on the machined surface. Due to the periodic nature of cutting forces in ball-end milling, vibrations in the tool and workpiece are unavoidable. Therefore, to predict the roughness of the machined surface, the effects of the dynamic deflections of the tool need to be taken into account.

Many different models have been proposed for simulating the structural dynamics of the machining systems. Among them, the modal analysis is a very practical method, which relies on the measured transfer functions of a structure. The depth of cut is small in ball-end milling. Hence, the transfer function of the cutting tool is measured at the tool tip, as shown in Figure 10. Only the structural dynamics of the tool is considered for simplicity. Therefore, when the workpiece and/or the machine tool do not have enough stiffness, this structural model may not be accurate enough. However, the dynamics of the workpiece and the machine tool can also be determined with the help of finite element analysis.

Figure 10.

Dynamic model of the tool in ball-end milling.

There may be several spindle and tool modes in the X_T and Y_T directions. Correspondingly, the dynamic cutting force can be related to the dynamic displacement of the tool by using the transfer function matrix as

$$H_x(s)=\frac{{\delta }_x(s)}{F_x(s)}=\sum _{i=1}^N\frac{1/m_{\mathrm{xi}}}{s^2+2{\zeta }_{\mathrm{xi}}{\omega }_{\mathrm{nxi}}s+{\omega }_{\mathrm{nxi}}^2}$$

(20)

where s is the Laplace operator; N is the number of modes in X_T direction of TCS and ω_nxi, ζ_xi and m_xi are the natural frequency, structural damping ratio and mass of mode i, respectively. Similar transformation can be used for structural displacements δ_y and cutting forces F_y. The transfer function can be converted into observable state-space canonical form, and then it can be solved by using the Runge–Kutta method. When the dynamic displacements (δ_x, δ_y) are obtained, ε _T,T in equation (13) can be replaced by (δ_x, δ_y, 0)^T. Then, the roughness of the machined surface can be evaluated.

Case studies

To validate the proposed methods, several simulations have been conducted in MATLAB R2007b. In this section, some results are presented and analyzed. The configuration of the machine tool used in the simulations is shown in Figure 2. The workpiece is a rectangular block of AISI 1050 steel. A two-flute ball-end mill was used. The diameter of the ball-end mill is 8 mm. The helix and radial rake angles are 30° and 8°, respectively. The overhang length of the cutter is 55 mm. The cross-feed direction is positive. Young’s modulus of the tool is 620 GPa. Cutting force coefficients are calculated using the mechanics of milling method. ⁶ The chip thickness ratio (r), friction angle (β, °) and shear stress (τ, MPa) are generated based on the uncut chip thickness ct (mm) and the cutting speed V (m/min) ¹¹

$$\begin{array}{c}\multicolumn{1}{c}{r=0.4+0.0005V+0.6\mathrm{ct}}\\ \multicolumn{1}{c}{\beta =26.8-0.0313V+11.77\mathrm{ct}}\\ \multicolumn{1}{c}{\tau =450.3+0.4V+227.5\mathrm{ct}}\end{array}$$

(21)

In those simulations, the ball-end mill moves along the X- and Y-axes of the machine tool. The geometrical errors of six equally spaced points along the two axes are generated randomly and listed in Table 2. The geometrical errors of other points can be obtained by using the interpolation method. Axes Z, A and B are fixed in the simulations. Their geometrical error components are listed in Table 3.

Table 2.

Geometrical errors of the X- and Y-axes.

No.	X-axis						Y-axis
	α	β	γ	Δx	Δy	Δz	α	β	γ	γS	Δx	Δy	Δz
	(×10−6 rad)			μm			(×10−6 rad)				μm
1	−1.5	1.5	−2	1.5	2	0.5	2.5	1	1	−5	2	−1	2
2	0.5	3	−4	3	0.5	2	4.5	0.7	−1	−3	1	−2	3
3	3.5	9	−7	4	−0.5	3	7.5	−0.5	−3.5	−2	−0.5	−4	4.5
4	2	7	−4	0.5	0.5	3.5	5	0.5	−4	−3	−1	−1	6
5	4	5	−2	1	1	4.5	6.5	1.3	−6	−1	−2	0.5	5
6	3	6	−4	3	1.2	4	8	1.5	−6.5	−2	−1.5	1.5	4.5

Table 3.

Geometrical error of the Z-, A- and B-axes.

Axis	α	β	γ	Δx	Δy	Δz	αS	βS	γS
Z	−3.5	−2.5	1.5	0.8	1.4	1.6	4	−3	0
A	1	−2	−2	1.5	0.5	0.5	3	−5	0
B	−2	−5	2	1	−1.5	1	2	0	−2

By using the method presented in section “Multi-scale modeling of the workpiece,” the datum of the machined workpiece is obtained. The locating errors are calculated by using the method presented in section “The effects of locating errors” (Figure 11). The obtained parameters of the locating errors are listed in Table 4.

Figure 11.

Calculation of the locating errors: (a) geometry of the datum of the workpiece and (b) convex hull of the difference face.

Table 4.

Parameters of the locating errors.

α (rad)	β (rad)	γ (rad)	Δx (mm)	Δy (mm)	Δz (mm)
9.0 × 10−4	2.2 × 10−4	−2.3 × 10−4	5.0 × 10−4	0.0456	0.0111

The modal data of the ball-end mill are shown in Table 5.

Table 5.

Modal data of the ball-end mill.

Mode No.	ωn (Hz)	ζ (×10−3)	m (kg)
1	1826.7	9.993	0.0418
2	1934.4	9.584	0.0824
3	2028.4	9.726	0.0852

Prediction and analysis of the form deviation of the machined surface

Four different cases are carried out to demonstrate the effects of the locating errors and geometrical errors of the machine tool on the form deviations of the machined surface. The simulated area and mesh size are 100 × 100 and 2.5 × 2.5 mm², respectively. The spindle speed is 12,500 r/min, the feed rate is 0.34 mm/tooth and the cutting depth is 1 mm. The results are shown in Figure 12.

Figure 12.

Analysis of the form deviations of the machined surface: (a) considering only the effects of tool deflections; (b) considering only the effects of locating errors, geometrical errors of machine tool and tool deflections; (c) difference values of the results with and without considering the effects of geometrical errors of the machine tool and (d) difference values of the results with and without considering the effects of locating errors.

In case 1, neither the locating errors nor the geometrical errors of the machine tool are considered in the simulation process; the predicted form deviations are shown in Figure 12(a). In case 2, both these factors are considered; the result is shown in Figure 12(b). In case 3, the geometrical errors of the machine tool are not considered, and the difference values of the results of cases 2 and 3 are shown in Figure 12(c). In case 4, the locating errors of the workpiece are not considered, and the difference values of the results of cases 2 and 4 are shown in Figure 12(d). From those results, the locating errors are up to 3.5 μm, and the geometrical errors of the machine tool are even close to 0.015 mm. Therefore, it is necessary to take those factors into account in the modeling of the milling process. From Figure 12(c) and (d), it can be found that the geometrical errors of the machine tool mainly affect the form deviations of the machined surface, and the locating errors of the workpiece affect the location deviations.

Prediction and analysis of the roughness feature of the machined surface

To make a more accurate presentation of the machined surface, the roughness feature is predicted and analyzed in this section. Several simulations are carried out to demonstrate the proposed method. The size of the simulated area is limited to 7.5 × 7.5 mm². The mesh size is 0.005 × 0.005 mm².

Figure 13 presents the predicted topography of the machined surface under the following cutting condition: a spindle speed of 14,500 r/min, a feed rate of 0.04 mm/tooth, a cutting depth of 0.5 mm and a stepover of 0.1 mm.

Figure 13.

The topography of the roughness feature at 14,500 r/min, 1 mm cutting depth, 0.1 mm stepover and 0.04 mm/tooth feed rate.

Because the effects of dynamic deflections of the tool are included in the model, the stability of the milling process can be analyzed. Figure 14(a) shows the predicted topography of a stable cutting process. Figure 14(b) shows the displacement spectrum of the tool; the frequency components concentrate at the tooth passing frequency ωt (416 Hz) and its multiples, which demonstrates that the process is stable. The machining errors along the feed direction (Figure 14(c)) also show that the vibrations of the tool are stable.

Figure 14.

Stable cutting at 12,500 r/min, 0.4 mm cutting depth, 0.1 mm stepover and 0.05 mm/tooth feed rate: (a) predicted topography of the machined surface, (b) displacement spectrum of the tool and (c) profile of the roughness feature in the feed direction.

For comparison, Figure 15 shows an unstable milling process. From Figure 15(b), besides the frequency components at the tooth passing frequencies, there is a frequency component at the chatter frequency (1847 Hz), which demonstrates that the process is unstable. The vibrations of the ball-end mill are more violent as shown in Figure 15(c).

Figure 15.

Unstable cutting at 12,500 r/min, 1.4 mm cutting depth, 0.1 mm stepover and 0.05 mm/tooth feed rate: (a) predicted topography of the machined surface, (b) displacement spectrum of the tool and (c) profile of the roughness feature in the feed direction.

Besides the stable and unstable cases, a series of simulations were carried out to analyze the effects of the cutting parameters on the roughness of the machined surface. The basic cutting conditions for those simulations are as follows: a spindle speed of 14,500 r/min, a feed rate of 0.05 mm/tooth, a stepover of 0.1 mm and a cutting depth of 0.5 mm.

Figure 16 shows the influences of the applied cutting depth. The analysis refers to three different groups of cutting parameters. From Figure 16(a)–(c), it can be found that despite the changes in feed rates, spindle speeds and stepovers, the roughness increases generally with the increase in the cutting depths.

Figure 16.

The effect of cutting depth at (a) different feed rates, (b) different spindle speeds and (c) different stepovers.

From Figure 17, it can be found that the influences of the spindle speed are irregular. It is mainly because the effects of the spindle speed are reflected by feed rate. For a fixed spindle speed, the feed rate will be different at different feed speed (mm/min).

Figure 17.

The effect of spindle speed at (a) different cutting depths, (b) different feed rates and (c) different stepovers.

Figure 18(b) presents the characteristic line of the cut remainder left by the feed movement of the tool. The height of the ridge can be written as

$$h_f=R_0-\sqrt{R_0^2-f_t^2/4}$$

(22)

Figure 18.

Analysis of the roughness of the surface machined by ball-end mill: (a) surface topography and (b) characteristic line in the feed direction.

In practice, the height of the ridge is much smaller than the radius of the ball-end mill. Therefore, equation (22) can be further simplified as

$$h_f=\frac{f_t^2}{8·R_0}$$

(23)

The height of the ridge left by the cross-feed movement of the tool can be obtained similarly. Furthermore, in the spherical-tool approximation model, the maximum surface roughness h_max is approximated by summing the heights of the ridges left by the feed and cross-feed movements of the tool as

$$h_{max}=\frac{f_t^2}{8·R_0}+\frac{s_t^2}{8·R_0}$$

(24)

The mean line of the roughness profile is h_max/8 from the bottom of the trough. Using the mean line, the roughness value is about one-quarter of h_max, that is ³⁸

$$R_a\approx \frac{h_{max}}{4}$$

(25)

From equations (24) and (25), it can be found that there is a quadratic relationship between the roughness value R_a and the feed rate used in the milling process. And that is the reason for the quadratic phenomenon shown in Figure 19(d).

Figure 19.

The effect of feed rate: (a) at different cutting depths, (b) at different spindle speeds, (c) at different stepovers and (d) quadratic fitting.

The movements of the tool in the cross-feed direction can be taken as the tool feed in the C direction with a feed rate as the stepover value. Therefore, the stepover shows similar effects on the roughness of the machined surface as the feed rate does (Figure 20).

Figure 20.

The effect of stepover at (a) different cutting depths, (b) different feed rates, (c) different spindle speeds and (d) quadratic fitting.

Based on equations (24) and (25), the R_a value is mainly determined by the feed rate and the stepover. In Figures 14 and 15, same feed rate and stepover are adopted. However, their results are quite different. When the oscillations occurred in the beginning of the simulations were ignored, the simulation result of the stable cutting case is very close to the value predicted by equation (25), which is about 0.3 μm. The simulation result of the unstable cutting case reaches 1 μm or three times the R_a value of the stable cutting case. The reason for this phenomenon is that the dynamic movements of the tool are not considered in equation (25).

To investigate the significance of the influence of cutting parameters on the quality of the machined surface, a group of orthogonal experiments were conducted. The experiments were designed as three levels for each factor, and their different combinations construct the standard L9 (3⁴) orthogonal arrays as shown in Table 6. Their corresponding R_a values were listed in the last array.

Table 6.

Results of the orthogonal experiments.

No.	Cutting depth (mm)	Spindle speed (r/min)	Feed rate (mm/tooth)	Stepover (mm)	Ra (μm)
1	0.5	12,500	0.03	0.1	0.0891
2	0.5	13,500	0.05	0.14	0.1783
3	0.5	14,500	0.07	0.18	0.3565
4	0.7	12,500	0.05	0.18	0.2849
5	0.7	13,500	0.07	0.1	0.1946
6	0.7	14,500	0.03	0.14	0.1817
7	1.0	12,500	0.07	0.14	0.3900
8	1.0	13,500	0.03	0.18	0.2661
9	1.0	14,500	0.05	0.1	0.1853

For each factors, the average R_a values at each level are listed in Table 7. Those values are also plotted in Figure 21, from which it can be found that the result ranges of the feed rate and stepover are larger than the cutting depth and spindle speed. Therefore, in the analyzed range, the feed rate and stepover have more significant influence on the roughness of the machined surface. Since the feed rate (feed per tooth) is determined by both the spindle and the feed speed, a suitable combination of them is important to the quality of the machined surface.

Table 7.

Average R_a values for each factors at each levels.

Factors	Level 1	Level 2	Level 3	Max–Min
Cutting depth	0.2080	0.2204	0.2805	0.0725
Spindle speed	0.2547	0.2130	0.2412	0.0417
Feed rate	0.1790	0.2162	0.3137	0.1347
Stepover	0.1563	0.2500	0.3025	0.1462

Figure 21.

Significance analysis of the cutting parameters.

Conclusion

A multi-scale simulation system for predicting the overall geometrical features of the machined surface is proposed in this article. The effects of locating errors, geometrical errors of the machine tool and tool deflections on the surface quality are incorporated into this model. In modeling of the milling process, an approach based on the convex hull theory is proposed to identify the locating errors; a new method for calculating the cutting forces and updating the machined surface is developed; a new strategy for multi-scale predicting the surface deviations of the machined workpiece is proposed. At the end of this article, case studies are carried out to validate the proposed method. From the results of those case studies, it can be found that the locating errors and geometrical errors of the machine tool have major effects on the form deviations; the cutting parameters affect both the form deviations and roughness of the machined surface. Since the effects of those factors can be predicted, proper cutting parameters and error compensation strategies can be determined to improve the accuracy and quality of the machined surface.

The model is made taking some assumptions, such as small error assumption, considering the workpiece and machine tool as rigid elements and so on. To an extent, those simplifications and assumptions constrain the scopes of applications of the developed model. For example, it is not valid for the compliant workpieces and machine tools with weak stiffness. And in the modeling of ball-end milling operations, only the effects of locating errors, geometrical errors of the machine tool and tool deflections on the quality of the machined surface are included. Although other types of errors are not considered in this article, interfaces to them are included. For example, the model of the geometrical errors of the machine tool (equations (9)–(11)) can be extended to include the thermal errors of the machine tool, the spindle errors and the setting errors of the tool. The effects of those factors will be pursued and reported in the future.