Study on Scanning Forming Methods of Machining Work-Piece Surface

: In the process of machine tool cutting, there are strict geometric relations among the cutting edge curve / tool surface, machine tool movement and workpiece surface, and the machine tool movement is also related to the type of tool. Firstly, the forming methods of cutting workpiece surface are analyzed and summarized from the geometric point of view, and the scanning forming method and its geometric expression are studied, and the research technical route of forming turning scanning forming is put forward. Then, the mathematical modeling and Simulation of forming turning are carried out according to the proposed technical route. Finally, taking the groove of the inner ring of the formed turning ball bearing as an example, the mathematical modeling of the design surface of the workpiece and the machined surface of the workpiece is carried out. The radial dimension changes of the workpiece caused by the cutting force and tool wear are analyzed, and the simulation of the machined surface of the workpiece is carried out.

there is a rigorous geometric relationship among the tool cutting edge curve/tool surface, the machine motion and the workpiece surface.
With the rapid development of society and economy, in order to meet the needs of product functions, more and more mechanical parts with complex curved surfaces are used on the surface, and the cutting and forming movements of machine tools are also becoming more complicated. In 1952, CNC machine tools [3,4] came into being. The cutting and forming process of CNC machine tool processing and ordinary machine tool processing mechanical parts is the same, only in the control method is different, there are also problems with the theory and method of part cutting and forming. Yuan-Liu Chen et al. [5] presented a self-evaluation method for submicron precision measurement of the cutting edge contour of a micro diamond tool with a force sensorintegrated fast tool servo (FS-FTS) on an ultra-precision lathe. Zhihuang Shen et al. [6] presented a digital graphic scanning (DGS) method based on computer scanning images to generate grinding profiles, avoiding the difficulties caused by complex contact line equations. Zhiyong Chang [7] formulated effective cutting edge equations to accurately calculate the dimensional errors and the surface roughness, providing a new approach for high precision CNC turning programming. Cheng Yajun et al. [8] innovated the processing technology of ball bearing and summarized the grooves of ball bearing turned by forming turning tool. M.L.Wu and Zhang KF et al. [9,10] proposed an evaluation method of interrupt shaping planning based on radial raceway contour change (monitoring point). S.J.Wang et al. [11] designed a three-axis computer numerical control (CNC) grinding machine for grinding micro-V-shaped groove arrays on hard and brittle materials. Material removal and machining strategy were studied for the grinding of V-grooves array. J-F Hsieh et al. [12] presented a simple yet comprehensive method for the design and machining of a Geneva indexing mechanism with curved slots. The kinematics model of grooved wheel mechanism was established by using homogeneous coordinate transformation and conjugate surface theory. Jui-Tang Tseng [13] presented the generative motion of machining curved-tooth cylindrical gear with CNC hobbing machine. Based on the cutting mechanism and the gear theory, the surface equation of this kind of gear is established, which is the function of the design parameters of the hob. Based on the mathematical model of the gear, the computer graph of the curved gear is given, and the tooth surface deviation caused by setting the nominal radius of the circular arc tooth line is studied. Yan He et al. [14,15] established a mathematical model and an analytical model based on tangential motion conditions, and deduced the profile of the rotary cutter according to the machining parameters and the required geometrical shape of the workpiece. The accuracy, calculation time and geometric flexibility are compared. Yingxue Yao et al. [16] analyzed and modeled the forming process of the workpiece in turning and the error sources affecting the machining accuracy of the workpiece, and presented to represent the geometric errors of the workpiece with the representation of the tool path and attitude. Hu Gongp et al.
[17] presented a new spiral tool path generation method for quasi-rotating diamond turning optical free-form surfaces based on spatial Archimedes spiral. Kun He et al. [18] used the forming tool to grind the spiral surface, calculated the point vector envelope method of the spiral surface forming tool contour, and verified the effectiveness of the method.
Based on the consideration of cutting tools classification, this paper analyzes and summarizes the forming methods of workpiece surface in cutting process from the Angle of geometry, and studies the scanning forming method among them. The mathematical modeling and simulation of forming turning were carried out by taking forming turning groove of ball bearing inner ring as an example. 2.Cutting shape of the workpiece surface 2.1Formation of the workpiece surface Based on the geometric angle, the cutting tools used in machining are divided into two categories. ① Non-rotating tools, such as turning tools, planers, etc., are geometrically embodied as cutting edge curves; ② Rotating tools, such as milling cutters, grinding wheels, etc., are geometrically embodied as the tool surface where the cutting edge of the tool is located.
Based on the geometric point of view, through the analysis and induction of the machining methods of the machine tool, there are the following two methods for the surface cutting and forming of the workpiece.
Use the first type of tool, that is, a non-rotating tool. The tool is embodied as a cutting edge curve, and the surface of the workpiece is formed by the motion scanning of the cutting edge curve of the tool.
Use the second type of tool, that is, the rotary tool, the tool is embodied as the tool surface where the cutting edge of the tool is located, and the surface of the workpiece is the envelope of the surface family formed by the movement of the tool surface.
When forming cutting, the cutting edge of the tool is in line contact with the surface of the workpiece. It belongs to the method of using the first type of tool to scan the surface of the formed workpiece with one pass. The machine tool only needs a single parameter movement, and the cutting efficiency is high.

2.2Formation of the workpiece surface
When using a shaped planer for planing processing, the curved translational movement of the cutting edge of the shaped planer forms the surface of the workpiece, as shown in Fig.1. Transform the cutting edge curve to form the surface of the workpiece. In order to ensure the tool angle during forming turning, the relative posture of the forming turning tool and the workpiece must be kept unchanged. Therefore, the workpiece revolving surface is formed by the circular trajectory of the cutting edge curve of the forming turning tool around the axis of the workpiece spindle, that is, the normal motion scanning of the cutting edge curve of the forming turning tool forms the surface of the workpiece [3].
As shown in Fig.2, the cutting edge curve Lt is rotated around the workpiece axis to obtain the workpiece surface Sp. Establish the tool coordinate system consolidated with the tool and the workpiece coordinate system consolidated with the workpiece, establish the mathematical model of the tool cutting edge curve in the tool coordinate system, and use the translation transformation matrix from the tool coordinate system to the workpiece coordinate system to convert the tool cutting edge curve Move to the workpiece coordinate system.
2. Mathematical modeling of workpiece design surface Using the rotation transformation matrix, a mathematical model of the workpiece design surface is formed from the cutting edge curve of the tool.
3. Mathematical modeling and simulation of the surface of the formed turning workpiece considering the relative displacement When cutting, factors such as cutting force and tool wear will cause the relative displacement between the tool and the workpiece, and then produce machining errors between the surface of the workpiece and the design surface. Consider the relative displacement between the tool and the workpiece, and use coordinate transformation to establish a mathematical model of the workpiece surface.

Research on influencing factors
Study the influence of factors such as cutting force and tool wear on the relative displacement between the tool and the workpiece.

3.Mathematical Modeling of Workpiece Design Surface for Forming Turning
As shown in Fig.3, establish a tool coordinate syste 0 0 0 0 0 ( ) with the tool. Select the point on the cutting edge curve of the forming turning tool that is closest to the axis of the machine tool spindle as the tool coordinate system origin O0, and the Z0O0X0 plane is coplanar with the workpiece rotation axis.

Fig. 3 Tool coordinate system
The parameter Eq.(1) for establishing the cutting edge curve Lt is as follows:

3.2Mathematical modeling and solution of design surface
As shown in Fig.4, establish a workpiece coordinate system 1 consolidated with the workpiece. The workpiece coordinate system is parallel to the tool coordinate system, and the Z axis coincides with the machine tool spindle. The X axis of the workpiece coordinate system coincides with the X axis of the tool coordinate system. The distance between the origins of the two coordinate systems on the X axis is r0, and r0 is the minimum radius of gyration of the workpiece surface. Transform the tool cutting edge curve Lt in the tool coordinate system to the workpiece coordinate system, the Homogeneous coordinate translation transformation matrix T is determined as Eq.(2).
In the formula, Tx is the X-axis coordinate of the tool coordinate system origin O0 in the workpiece coordinate system, Tx=-r0. After translation transformation, the parameter Eq.(3) of the cutting edge curve in the workpiece coordinate system is obtained.
The homogeneous coordinate rotation transformation matrix RZ is Eq.(4).
After the rotation transformation, the parameter Eq.(5) of the workpiece design surface Sp in the workpiece coordinate system is obtained.
In the tool coordinate system, the coordinate of any point Pe on the cutting edge curve is       Taking forming turning tool turning standard GB/T 276-2013 6202 deep groove ball bearing inner ring groove as an example, the design surface of the workpiece is solved. Fig.5 shows the inner ring dimensions of 6202 deep groove ball bearings.
Substituting Eq.(9) into Eq.(8) and solving with Matlab, the design surface shown in Fig.7 is obtained. In the cutting process, factors such as cutting force and tool wear will cause relative displacement between the tool and the workpiece, which will result in machining errors. In the X and Y directions, the relative displacement between the tool and the workpiece is expressed as δX and δY, respectively. In the tool coordinate system, the parameter Eq.(10) of the cutting edge curve considering the relative displacement between the tool and the workpiece is established.
After translation transformation, the parameter Eq.(11) of the cutting edge curve in the workpiece coordinate system considering the relative displacement between the tool and the workpiece is established After rotating transformation, the workpiece surface Eq. (12) in the workpiece coordinate system considering the relative displacement between the tool and the workpiece is obtained:

Fig. 8 The influence of X-direction cutting force
The surface of the workpiece considering the influence of the X-direction cutting force is shown in Fig.8. The relative displacement between the tool and the workpiece under the action of the X-direction cutting force is expressed as δfX, and the parameter Eq.(13) of the cutting edge curve in the tool coordinate system is established as follows: x u After translation transformation and rotation transformation, the workpiece surface parameter Eq. (14) in the workpiece coordinate system considering the influence of the X-direction cutting force can be obtained. Fig. 9 The influence of Y-direction cutting force Fig.9 shows the workpiece surface considering the influence of Y-direction cutting force. The relative displacement between the tool and the workpiece under the action of the Y-direction cutting force is expressed as δfY, and the parameter Eq.(15) of the cutting edge curve in the tool coordinate system is established as follows: After translation transformation and rotation transformation, the workpiece surface parameter Eq. (16) in the workpiece coordinate system considering the influence of the Y-direction cutting force can be obtained.
3. Workpiece surface modeling considering the comprehensive influence of X and Y cutting forces The surface of the workpiece considering the comprehensive influence of the X-direction Y cutting force is shown in Fig.10. The parameter Eq.(17) of the cutting edge curve in the tool coordinate system is established.
As shown in Fig.11, from the perspective of machining accuracy, the tool wear NB along the radial direction of the workpiece is generally used as a measure of bluntness during turning. The parameter Eq. (20) of the cutting edge curve in the tool coordinate system considering the influence of tool wear is established as follows: After translation transformation and rotation transformation, the workpiece surface parameter Eq.(21) in the workpiece coordinate system considering the influence of tool wear can be obtained.

Surface analysis of forming turning workpiece
Under the action of X-direction cutting force, the radius of the workpiece is given by, and δfX is selected as 0.005mm in this paper.
Using Matlab software to solve the groove surface of Eq.(5) and Eq. (14), the deviation between the workpiece surface and the design surface produced by the X-direction cutting force is shown in Fig.12. , and δfY is selected as 0.01mm in this paper.
Using Matlab software to solve the groove surface of Eq.(5) and Eq. (16), the deviation between the workpiece surface and the design surface produced by the Y-direction cutting force is shown in Fig.13. Using Matlab software to solve the groove surface of Eq.(5) and Eq. (18), the deviation between the workpiece surface and the design surface produced by the comprehensive influence of the X and Y cutting forces is shown in Fig.14. Because the linear velocity of each point on the cutting edge is different, the amount of tool wear at each point on the cutting edge is also different. In this paper, the relationship between tool wear and cutting speed is simplified as a proportional relationship.
Due to tool wear, the radius of the workpiece is changed by v rr  , 2 2 v aX aY rr     ( )

Figure 1
Shaped and planed curved surface   Application of Matlab to solve the design surface The in uence of X-direction cutting force The in uence of Y-direction cutting force Figure 10 Comprehensive in uence of X and Y cutting force Figure 11 Tool wear Figure 12 The in uence of X-direction cutting force Figure 13 The in uence of Y-direction cutting force Figure 14 Comprehensive in uence of X and Y cutting force Figure 15 The effect of tool wear