Modelling and analysis of surface topography generated in end milling process

The surface topography of workpiece plays an important role in the performance and service life of workpiece. Complex surface parts are widely used in shipbuilding, aerospace and other industries. At present, the study of milling surface topography is mainly on 3-axis milling. A prediction model of milling surface topography is proposed, which can obtain the machined workpiece surface topography and roughness directly from cutting parameters, cutter location file and workpiece surface geometry. The effects of cutting parameters on surface roughness is discussed. Different milling experimental conditions are set up to validate the proposed model. This method can be used to analyze the surface topography of milling, and further to optimize the cutting parameters to improve the surface quality.


Introduction
The surface topography has an important impact on the performance and fatigue of the workpiece [1,2]. The prediction of milling surface topography is necessary, which is the theoretical basis of cutting parameter optimization. With the development of intelligent manufacturing, data processing method is used to analyze cutting characteristics, such as cutting force, cutting temperature and surface topography. Khorasani et al. [3] used artificial neural networks method to analysis the surface roughness, and the data is obtained by designed experiments. Liu et al. [4] used multi-sensor information to reconstruction of 3D surface topography modeling. Ngerntong et al. [5] used Taguchi method and experimental results to analysis and modeling the machined surface roughness. Chen et al. [6] designed orthogonal regression experiment method to predict the micro-milling surface roughness. In previous works [3][4][5][6], the surface topography and surface roughness prediction model can be obtained by analyzing and processing the experimental data.
This method is used in specific material and cutting parameters, which is impossible to analyze the formation process of milling surface topography theoretically.
By analyzing the motion characteristics of milling process, the surface residual height after machining forms surface topography. The geometry of complex surface is difficult to be expressed analytically, and the discrete method is usually used, such as z-map method, z-buffer method. Z-map method is used to calculation the tool-workpiece engagement in ball end milling [7]. Gao et al. [8] used z-map method to simulate the machined surface topography and roughness in ball end milling processes. Similarly, Zhang et al. [9] developed a improved z-map method to modeling the surface topography in ball end milling. Based on the proposed model, the effect of cutting parameters on the surface topography is discussed. B. Denkena et al. [10] simulate the surface topography in five-axis milling and the tool's dynamic excitation is considered. Peng et al. [11] developed the surface topography based on the cutting edge motion trajectory model, and the effect of inclination angle is discussed. Liu et al. [12] developed the prediction model of the surface topography in ball screw whirling milling. Zhang et al. [13]proposed a new surface topography model which considered the cutter deflection in micro-side milling. Kasim et al. [14] used experimental results to analysis the surface topography in ball nose end milling process, the difficult-to cut materials and tool wear are considered in the model. Arizmendi et al. [15] proposed the model of face milling surface topography based on the cutting edge geometry and trajectory. Wang et al. [16] developed 3D surface topography of ruled surface milling, and the complex workpiece surface is described by point cloud. Xu et al. [17] present a surface topography model for a ball-end cutter, and the feed rate is considered in high speed milling. Torta M et al. [18] developed a framework model for estimating the surface texture in complex surface milling process.
The above studies is mainly on ball-end mill and 3-axis milling. The study of surface topography model with end mill is universal and necessary. The modeling of milling surface topography is introduced in Section 2. In order to analysis the effect of cutting parameters on surface topography and roughness, and validate the proposed model, experiment and results are discussed in Section 3. In section 4, the conclusion of this study is summarized.

Theoretic modeling of surface topography
In the milling process, the residual height in the feed direction and intermittent feed direction after milling will have an impact on the machined surface topography. Therefore, it is necessary to establish the point cloud swept by the cutting edge in the milling process of end milling according to the geometric model of milling cutter, milling process kinematics and tool path file in the milling process. The point cloud formed in the process updates the discrete points of the workpiece surface, and finally obtains the machined surface topography.

Geometry model of the torus end mill
The geometric model of torus end mill is shown in Fig.1. D is diameter, r is the bottom radius, () rz is the filleted radius, and point O T is the tool center. As shown in Fig. 1c, the effective cutting radius of point P can be calculated based on the axial position: For circular inserted cutter, the helix angle of the cutting edge of the arc segment is not a fixed value .The helix angle of element P can be expressed as the following equations: where 0 i is maximum helix angle.
The radial lag angle () z  and axial immersion angle () z  as follows: ( ) cos , 0 rz z arc z r r The radial immersion angle of point P at level z can be calculated: ( , ) ( where is tool rotation angle and N is cutter edge number.

2.2
The movement track of discrete points of cutting edge By analyzing the influence of tool size, tool movement relative to workpiece, tool inclination angle and different cutting edges on the trajectory of discrete points, the trajectory equation of each discrete point in the workpiece coordinate system in the machining process is established. The movement track of cutting edge of torus-end milling on the m-th cutting path is shown in Fig.2. Fig. 2a shows the projection of the motion track of the point on the cutting edge on the second cutting path on the plane. Because the points on the cutting edge rotate around the cutter axis and move along the feed direction in the milling process, the trajectory of the points on the cutting edge is asymmetric. When the torus end mill moves along the feed direction of the first cutting path, the points on the cutting edges 1 and 2 will sweep away the material on the workpiece. Suppose that in the process of cutting, the cutting edge 1 cuts the workpiece before the cutting edge 2, and the residual material after cutting the cutting edge 1 will be cut again by the cutting edge 2 (as shown in Fig.2b). After cutting in the first cutting path, the residual height on the workpiece surface will be cut off by the cutting edge in the next cutting path. Therefore, residual height in the feed direction is formed by the sweeping tracks of four cutting edges( The ij Z coordinate of each meshing point represents the height of the workpiece : A set of coordinate systems are established, as shown in Fig. 5.
represents a workpiece coordinate system, represents local tool coordinate system, In the simulation, the cutter was discretized into k disks, the ( , , ) c c c x y z at k disks at local coordinate system can expressed by the following: In the proposed method, the tool orientation and position are obtained from the CL file. The cutter tip orientation and position in the tool path files is represent as ( , , , , , ) x y z    in the O XYZ  system, the ( , , ) x y z represent the position and the ( , , )    represent the orientation. The tool axis vector ( , , ) xt yt zt is: Consequently, the overall transformation matrix M from is written as: The general form of the trajectory equation of point P in Assuming that the height of the point on workpiece surface corresponding to ( , ) ij XY is ij Z , the scallop height of z after material removal is following: The roughness is the arithmetic mean deviation of contour a R 1 1 () n at R t t n   (15) As shown in Fig.6, in the whole milling profile simulation system, tool path file (including tool position and posture), spindle speed and feed speed, geometric parameters of tool and workpiece are included. In the preparation stage of simulation, the coordinates of discrete points are used to represent the workpiece surface. The next step is to use these input parameters for simulation. In the simulation process, the main task is to check the tool workpiece contact, that is, to determine whether the cutting edge of the tool has cut to the workpiece surface. If the workpiece has been cut to the cutting edge, the Z value of the grid point needs to be updated to the surface height value after cutting. Repeat the above process until the whole workpiece has been cut, and the final calculation will be made. The specific calculation steps are as follows: (1) Input the tool path, cutter location file, the tool geometry parameters, workpiece surface geometry.
(2) Divide the workpiece in plane, and discrete point file stored in the initialization matrix.
(3) Calculated the discrete points on the cutting edge in the workpiece coordinate system.
To validate the surface topography model, the cutting parameters is set to obtain large roughness for observation obvious.It is carried out on a block of 100 mm length, 10mm width and 10 mm height, which is divided into seven areas in the length direction and processed with different processing parameters. After all milling is completed, the three-dimensional surface topography is measured by the keinsys vh-m100 micro measurement system. The keinsys ultra depth of field micro measurement system can reconstruct, display and measure the undulation of the surface. Therefore, the images before and after the experimental processing can be easily compared, and the effectiveness of the simulation   Table 1 is the different spindle speed in milling. Ra is the roughness of the surface which is measured by instrument. In Table1, the surface roughness of the workpiece after cutting under different spindle speeds parameters is shown. Fig.8 is the feed direction roughness with different spindle speed. The feed direction roughness is decrease with the spindle speed increase. Since the spindle speed increase, intermittent cutting time between the two cutting edges decrease, the more conducive to the formation of relatively small surface residual height.   Table 2 is the different feed speed in milling. Fig.9 is the feed direction roughness with different feed speed. As shown in Fig.9, the roughness value in the milling feed direction is increasing with the increase of milling feed speed. But in the actual production, the lower the feed rate is not the better, because in the actual production, not only the surface quality of the workpiece after processing is concerned, but also the processing efficiency is concerned. Choosing a lower milling feed rate is conducive to improving the surface quality, but it will reduce the production efficiency.  Fig.9 Relationship between speed speed and feed direction roughness Table 3 is the different lead angle milling tests and the roughness in feed direction. Fig.10 is the surface roughness results after milling with the different incline angle. When the tool inclination angle is 0 degree, the cutter tip velocity is zero, which lead to the poor surface quality. A suitable tool inclination can improved the tool life, surface quality significantly. The roughness decreases rapidly with tool inclination angle increase. And when the tool inclination angle increased to 15 degree, roughness changed slowly. When the inclination angle is increase to 20deg, the roughness increase. In the milling process, the tool elastic deformation is increase with the tool-surface inclination increase. An inclined angle of 10 degree to15 degree is suitable to obtain a good surface quality after milling.

Surface topography and scallop height validation
For comparison, the measured surface topography measured by the optical microscope is shown in Fig.12. From   Fig.12, it can be seen that on the surface of the workpiece after milling, the residual height of the workpiece in two directions is left after milling by ball end milling cutter. From the simulation results in Fig.12 b,c, it can be seen that the simulation program can clearly show the residual height of the workpiece in two directions after milling and the residual height of the workpiece surface after milling.

Conclusions
An analytical model is proposed for the prediction surface topography in end milling . The surface topography calculated using z-map method directly from CAM data. The tool runout, tool deformation, tool wear and vibration are not considered in this study. The following conclusions can be summarized: (1) The proposed model is used to predict surface topography directly from the CAM data and cutting parameters.
(2) The effect of cutting parameters(spindle speed, feed speed, inclination angle) on surface topography is discussed.
(3) The comprehensive experiments are used to validate the model, and the results of surface roughness is consistent and have certain errors between measured and simulation.

Funding information
The authors would like to thank the National Key Basic Research Program (NKBRP) of China (No. 2014CB046704), the Starting Research Fund from the Hubei University of Arts and Science.

Authors' contributions
Ruihu Zhou designed and performed the manuscript, analyzed the data, and drafted the manuscript. Qilin Chen designed and carried out the experiments. All authors have read and agreed to the published version of the manuscript.
Ethical approval Not applicable.

Consent to participate Not applicable.
Consent to publication All presentations of case reports have consent for publication.
Availability of data and material All data generated or analyzed during this study are included in this published article.

Conflict of interest
The authors declare they have no conflict of interest.