A novel grinding method for the flute with the complex edge by standard wheel

The flute plays an important role in the design process for tools. The complexity of the cutting edge is increasing continuously. Thus, it is difficult to grind flute precisely and efficiently. This paper presents a practical grinding method for the flute of tools. It solves the issue of how to grind the flute with the complex edge by standard wheels. It is summarized as follows: the curve of cutting edge and core are used as the designed parameters of the flute; we propose the fundamental conditions that the wheel does not interfere with designed parameters; we can find the posture and position of wheel in the curve frame easily, thus obtaining wheel trajectory. Finally, using a taper end mill as an example, the results show a good agreement between the CAM data and the model data. This proves the effectiveness of this method.


Introduction
Nowadays, the manufacturing industry is developing rapidly. Cutting tools are widely used, especially in aerospace, automotive, and mold. The flute is the key structure of tool. It directly affects the cutting performance and the tool life. For instance, the flute is strongly related to cutting forces, chip formation, and fracture. The core affects the rigidity of the end mill. Normally, the flute is generated by a wheel rotating around the axis of the blank, which is a complex surface. The efficient and accurate method of CNC grinding for the flute has attracted considerable critical attention in the industry. Currently, the design methods of flute are classified into the direct method and the indirect method. In the indirect method, the cross-sectional shape of flute is given, and the forming grinding wheel for machining the target flute is obtained by applying the envelope theory. In the direct method, the flute is machined by managing the position of standard grinding wheel in the tool coordinate system (Fig. 1).
Meshing theory is the basis of the indirect method. Many researchers have explored the following two topics. How to describe the profile of flute by equations? How to find the shape of wheel? Ehmann et al. [1] proposed a purely analytical approach, based on the common normal vectors at the points of contact that must intersect the axis of wheel, to calculate the profile of wheel. Li [2] discussed the relationship between the interpolation methods and the precision of flute fitted curve. Sometimes, the profile of flute is generated by the combination of non-enveloped and enveloped curves. Meshing theory cannot work smoothly. Then, the Boolean method was brought into the indirect method [3]. Based on particle swarm optimization, an automatic method of searching for a wheel position in flute grinding for a given shape of the helical flute and grinding wheel profile was presented [4]. In practice, the indirect method is decided that each flute must match a forming wheel. The profile of wheel must be accurate. So, these decrease the flexibility and efficiency of the indirect method. So far, grinding complex cutting edge curves is still a challenge by indirect method.
The direct method applies a known profile grinding wheel to grind the target flute. Zhang et al. [5] presented a direct method for calculating the profile of flute and the analytical expressions of flute were determined. Nguyen and Kho [6] analyze the relationship between the singular point of wheel and flute. Under knowing wheel, rake, and core conditions, Li et al. [7] developed an automatic search algorithm for finding the position of wheel. Based on the theories of analytic geometry and envelope, a method of five-axis flute grinding in cylindrical end-mills was presented, which ensures the core, rake, and flute width [8]. To make flutes of more complex shape, standard wheels of complex shape (e.g., 1B1, 1E1, 1F1, and 4Y1 wheels) were used to grind the flute [9]. Currently, precision manufacturing poses higher demands on cutting tools, the profile of flute, and complex cutting edge especially. However, the above articles do not consider the complex cutting edge in their models. It is hard to grind tools with complex cutting edge, such as a taper cylinder tool with constant helix. A method for grinding flute with complex edge by the edge of wheel has been reported [10], but the edge of wheel is worn easily. Chen and Bin [11] demonstrated the grinding model for the rake face of the taper ball-end mill with a CBN spherical grinding wheel, but the spherical grinding wheel is not a standard wheel. Thus, these methods are not less prevalent. In the factory, most flutes were grinding by the standard wheel (Fig. 3). It can be reshaped easily because the profile of standard wheel is simple.
According to the discussion, this paper proposes a direct method for grinding flute by the standard wheel. First, we build the model of cutting edge and the moving frame attached to the edge. Then, we established the model and equation of the core. After that, we presented a method for calculating the position of wheel. Finally, the effectiveness of the model is verified by a CAM example.

The cutting edge and the frame
In this paper, we take a taper end mill with constant helix angle as an example to explain this model. As shown in Fig. 2, O − XYZ is the global coordinate system, and the Z-axis coincides with the axis of the tool. In this system, the edge is expressed as where the angle θ is the circumferential angle of the projection of point in XOY plane. R tool (z) is the radius of tool, being a function of z. It is expressed as where the angle t is the taper of tool. R t is the radius of tool at z = 0. The helix angle of cutting edge [10] is the angle Fig. 1 The method of grinding flute between the tangent of edge and the tangent of the generatrix that generates the tool rotating surface. Thus, is given by where is the helix angle. The cutting edge with constant helix angle is given by The space curve frame is essential for this paper. It can describe the relative position conveniently. Take any point P i on the cutting edge, we establish a frame P i − T i B i N i at this point. In global coordinate system, P i is given by T i is the tangent vector of the cutting edge at P i point. N i is the normal vector of the tool rotating surface at P i point. We define the binormal vector B i as the vector product of T i and N i .
According to the relationship between the global coordinate system and the frame P i − T i B i N i , we can get the rotational matrix M R and the translational matrix M M.

The model of core
Usually, the core also belongs to a rotating surface and the generatrix is a line, for example, the cone surface and the cylinder surface. So, the surface of core is represented in global coordinate system by the equations where the angle c is the circumferential angle of the projection of point in the XOY plane. R core (z) is the radius of core, being a function of z . It is expressed as where the angle c is the taper of core. R c is the radius of core at z = 0.

The coordinate system and model of wheel
To represent the wheel revolving surface in a parametric form, a coordinate system is established. As shown in Fig. 3, O W − X W Y W Z W is the coordinate system of wheel, and the Z-axis coincides with the axis of wheel. In this system, the surface of wheel is expressed as where the angle w is the circumferential angle of the projection of point in X W O W Y W plane. R wheel (z) is the radius of wheel, being a function of z.

Generation of the grinding path
We illustrate the method of calculating the grinding path. During the grinding process of flute, the motion of wheel is continuous. However, the numerical control system needs to input the discrete points of the path. The path should be discrete when we calculate the path. Now, there are many discrete methods, such as the equal chord length method and equal parameter method. In this paper, the equal parameter method is used to discretize the cutting edge. Then, the position of the grinding wheel is calculated for each discrete point along the cutting edge. So, the grinding path consisted of the position of wheel in order. The procedure of the method that calculates the position of wheel is as follows: Step 1: Select any point on the round chamfer of wheel; we define this point as the grinding point G. G can be represented in the coordinate system of wheel by equation (Fig. 4).
The angle μ is the angle between the normal vector of G and the axis of wheel, is a constant, and 0 ≤ ≤ 2 . The initial position of wheel is shown in Fig. 4. This position must satisfy the following requirements: Step 2: The moving frame P i − T i B i N i and wheel are rotating around T i , and the rotational angle is . Normally, is set by users and affects the rake angle. We obtain a new moving frame P i − T i1 B i1 N i1 . Like step 1, we can get the rotational matrix M R2 , which is given by Step 3: As shown in Fig. 4, the moving frame P i − T i1 B i1 N i1 and wheel are rotated about B i1 with the rotational angle . The new moving frame is defined as P i − T i2 B i2 N i2 . At this position, the surface of wheel must be tangent to the surface of core theoretically. However, the minimum value of distance between two surfaces must be within the manual setting error in practice. Similarly, the rotational matrix M R3 is represented as This position of wheel (Fig. 4) satisfies the design requirements. It is obvious that the surface of wheel in global coordinate system is represented by the equation The position of wheel is represented by the equation The axis of wheel is represented by the equation Figure 5 is the principle of calculating the grinding path.

Numerical example and discussion
To illustrate the method, a CAM example of grinding flute was performed.

Parameters and calculation
A taper end mill is used in this example; Table 1 lists the parameters of tool. The flute is ground by the standard wheel which type is 1L1, and parameters are shown in Table 2. Now, we calculate the grinding path. The cutting edge is divided into discrete points of which the distance of points is 0.05 mm along the Z-axis. The error of core is ±0.01mm . The following are the grinding parameters: Then, we can calculate the positions and axis of wheel by the above data (Table 3).

Result and discussion
In this paper, we have two ways to find the profile of flute (Fig. 6). In the first way, the intersections of plane z with the surface of wheel that is located in the calculated positions are the profile of flute, where z is an arbitrary value (Fig. 7). The profile of flute can be achieved by μ = 10 • , γ = 10 • MATLAB's boundary algorithm. In the second way, the positions of wheel are imported into NUMROTO which is the center of competence for tool grinding. Then, we can get the result of CAM by NUMROTO-3D (Fig. 6). After that, we can export the profile of flute. Comparing the two  results of CAM, the line profile is within 0.06 mm. In Z = 5 and Z = 10 planes, the normal deviation along the profile of flute is shown in Fig. 8. The error may come from the accuracy of the grinding wheel discrete. It means that the CAM is accurate. Based on the profile of flute, we can obtain the core and the cutting edge easily. The theoretical and simulated radius of core are shown in Table 4. All errors satisfy the manual setting error. Table 5 is the position of the edge. The maximum position is 0.004 mm within a reasonable range.

Conclusion
In this paper, a novel and practical method for grinding flute is presented. This method realizes design parameters (the cutting edge and core) of flute that are ground by the standard wheel. A CAM example was performed to verify this method. The result of CAM exhibits that the parameters of the simulation of flute accord with the desired flute. Hence, this method can be applied to manufacturing of cutting tools with complex edge and variable radius of core. In addition, under the same condition of the same wheel, the same cutting edge, and the same core, we can adjust the parameters μ and γ to obtain the different flutes. By shifting the equation of the cutting edge curve, we can easily extend this method to the manufacturing models of tools and structures.
Author contribution HW: methodology, conceptualization, validation, writing-original draft, term, data curation. XL: funding acquisition, writing-review and editing. CY: writing-review and editing. SW: writing-review and editing, data curation. GZ: writing-review and editing, data curation. KZ: writing-review and editing, data curation.
Funding This research was funded by Projects of International Cooperation and Exchanges NSFC (51720105009).

Availability of data and materials
The datasets used or analyzed during the current study are available from the corresponding author on reasonable request.

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Ethics approval The content studied in this article belongs to the field of metal processing and does not involve humans and animals. This article strictly follows the accepted principles of ethical and professional conduct.

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