Mathematical Model and Tool Path Calculation for Helical Groove Whirling

The helical groove shape plays a key role in ensuring the adequate flute space of many screw components. In many situations, the helical groove is machined through profiled grooving cutter, which brings a huge cost. This study establishes the mathematical model of helical groove based on cross-section and presents an approach to calculate tool path using the whirling process which machines helical groove through enwrapping movements with standard cutters. Finally, a case study and the error analysis are provided to illustrate the validity of the developed models and algorithms, which offers an alternative method for further computer aided manufacturing.


INTRODUCTION
Helical grooves are of importance in many screw components, especially those which form the flutes like screw shafts and extrusion screws. Generally, the process of helical groove machining is that a helical swept groove is obtained based on a cylindrical workpiece using a disk-type tool. Extensive work on helical groove has been carried out. presented a mathematical model and sensitivity analysis for helical groove machining. Ivanov and (1998) provided a generalized analytical method for the profiling of rotation tools for forming helical grooves. Sun et al. (2008) presented a new simulation model for generation of the helical surface profiles on cutting tools. Zhang et al. (2006) presented a practical method of modeling and simulation for drill fluting. et al. (2010) proposed to use Bezier polynomials in approximating a cylindrical helical surface with constant pitch and in-plane generatrix known in discrete form. Wang et al. (2006) proposed the use of standard cutters (cutting blades), through whose enwrapping movements to produce the helical groove.
This study aims at presenting the mathematical model and the approach of calculating tool path for whirling the helical groove.

HELICAL GROOVE MATHEMATICAL
The helical groove can be considered as the result of the helical movement of a profile cur the curve, as the generatrix of the helical groove, can be chosen arbitrarily. In practice however, a particular section profile is commonly used. As illustrated in Fig. 1, suppose the helical groove is formed by the helical motion of the cross section profile along the z The helical groove shape plays a key role in ensuring the adequate flute space of many screw In many situations, the helical groove is machined through profiled grooving cutter, which brings a This study establishes the mathematical model of helical groove based on cross-section and presents an approach to calculate tool path using the whirling process which machines helical groove through enwrapping movements with standard cutters. Finally, a case study and the error analysis are provided to illustrate the validity of the developed models and algorithms, which offers an alternative method for further computer aided manufacturing Helical groove, toolpath calculation, whirling process Helical grooves are of importance in many screw cially those which form the flutes like screw shafts and extrusion screws. Generally, the process of helical groove machining is that a helical swept groove is obtained based on a cylindrical type tool. Extensive work on ve has been carried out. Jung-Fa (2006) presented a mathematical model and sensitivity analysis for helical groove machining. Ivanov and Nankov (1998) provided a generalized analytical method for the profiling of rotation tools for forming helical grooves.
. (2008) presented a new simulation model for generation of the helical surface profiles on cutting ted a practical method for drill fluting. Onacea . (2010) proposed to use Bezier polynomials in approximating a cylindrical helical surface with plane generatrix known in discrete (2006) proposed the use of standard cutters (cutting blades), through whose enwrapping movements to produce the helical groove.
This study aims at presenting the mathematical model and the approach of calculating tool path for

MATHEMATICAL MODEL
The helical groove can be considered as the result of the helical movement of a profile curve. In principle, the curve, as the generatrix of the helical groove, can be chosen arbitrarily. In practice however, a particular commonly used. As illustrated in Fig. 1, suppose the helical groove is formed by the cross section profile along the z Fig. 1: Example of helical groove and its generatrix axis in a cartesian coordinate system versors are i, j, k, then the curve Г can be expressed as: where, u is a variable parameter. In this case the equation of the cylindrical helical groove can be written as: where, θ is the rotation angle around where p z is the lead. For right handed helical grooves, is positive; otherwise, p is negative.

WHIRLING PROCESS OF HELCIAL
Whirling is an efficient process for producing helical grooves on three-axis CNC machines. It is a particular type of milling that the tool disk rotates at high speeds around a slowly turning

Mathematical Model and Tool Path Calculation for Helical Groove Whirling
China The helical groove shape plays a key role in ensuring the adequate flute space of many screw In many situations, the helical groove is machined through profiled grooving cutter, which brings a section and presents an approach to calculate tool path using the whirling process which machines helical groove through enwrapping case study and the error analysis are provided to illustrate the validity of the developed models and algorithms, which offers an alternative method for further computer aided manufacturing.
is a variable parameter. In this case the equation of the cylindrical helical groove can be written is the rotation angle around z axis, ‫||‬ = ଶగ , is the lead. For right handed helical grooves, p is negative.

HELCIAL GROOVE
Whirling is an efficient process for producing axis CNC machines. It is a particular type of milling that the tool disk rotates at turning workpiece. As There is a distance e between y and Y and an installation angle δ between z and Z. In machining, the tool disk rotates at a high speed making the cutting movement, in the mean time it moves along both z and x axes. For a discrete screw shaft, the whirling process is usually conducted using specially profiled cutters that fit in with the grooves, with the feed rate being linked to the screw lead. While this is the typical method of thread whirling, it is also feasible to cut the helical groove into shape with standard cutters as long as their tool paths are well planned as to enwrap the desired surface. The latter method provides much flexibility for the whirling process as well as for its implementation. It is recognized there are still different ways to create the helical groove in terms of the machining strategies and tool path. In this study however, we focus on tipped cutters (tool bits with nose radius less than 1.5 mm) and choose to use the cross section profile as the envelope of the cutter locations. In the process, cutting tools are supposed to sweep along the workpiece while moving in and out according to its helical groove. In this case the tool path is a special-shaped helix, which needs three-axis (C, X, Z) linkage to realize the enveloping process.

TOOL PATH CALCULATION MODEL
Meshing motion model between workpiece and cutting tool: In machining the helical grooves, the surface of cutting tool is tangential to the surface of helical groove at any arbitrary contact point to ensure no interference. Furthermore, there is a common tangent plane and normal on each tangent point.
According to Eq. 1 and 2 and for simplicity, the equation of cross section profile can be rewritten as: Similarly, the equation of helical groove can be expressed as: where, x and θ are two values.
Due to the common normal vector of workpiece and cutting tool, there is: Besides, in terms of the aforesaid equations, N (x N , y N , z N ) can be determined as: where, r is the radius of tool tip and |݊| can be expressed as: 2 2 (cos ( )sin ) (1 ( )) (sin ( )cos ) Cutter location data calculation: As shown in Fig. 3, point O (x T , y T , z T ) is the central point of tool disk and point N (x N , y N , z N ) is the central point of cutting tool, we can have: where, R is the radius of tool disk. While point P is on X axis and point N is on XOY plane, so ܰ ሬ ሬറ = ܲܰ ሬሬሬሬሬሬറ is on XOY plane, moreover, ܴ ሬറ = ܱܰ ሬሬሬሬሬሬറ also is on XOY plane, so in this case, the direction of ܴ ሬറ × ܰ ሬ ሬറ is same with the positive direction of Z axis. By assuming that the versor of the positive direction of Z axis is ݇′ ሬሬሬറ , there is: According to the process of whirling, there is an installation angleδ between z and Z axes, we have: (16) Considering the Eq. 9, 10, 15 and 16, there is: Figure 3 also shows that the vector ܰ ሬ ሬറ ×ܴ ሬറ is perpendicular to vector ‫ݎ‬ റ×ܴ ሬറ and it can be expressed as: where, the coordinates of ‫ݎ‬ റ, ܴ ሬറ , ܰ ሬ ሬറ can be obtained by Eq. 8,9 and 10. In this way, the central point of tool disk O (x T , y T , z T ) can be calculated by Eq. 14, 17 and 18. Furthermore, the numerical control points of machine tools can be expressed as: where, A is the coordinate of X in machine tools and C is the rotation angle of the work piece.

CASE STUDY
An example part:When whirling the helical groove, the work piece rotates around C axis and the tool disk moves along X axis and Z axis simultaneously. An example is given as follows.
The example part is a double-thread screw shaft. Its lead pz is 88 mm, outer diameter D is 85 mm, root diameter d is 55 mm, tool tip radius r is 1.2 mm and the cross section is determined by sampled points as shown in Table 1.
According to sampled points, cubic spline fitting is used to obtain a complete profile of the cross section and the profile (curve a) is shown in Fig. 4.   In this whirling, the tool disk radius R is 120mm and installation angle δ is 10°, Fig. 4 (curve b) demonstrates part of the theoretical cross section profile when θ is pi/6 and part of the data points are shown in Table 2. In terms of Eq. 14, 17 and 18, just as Table 2 shows part of data points in cross section, Table 3 shows the corresponding central point of tool disk O (x T , y T , z T ). Then, in terms of Eq. 19, part of cutter location data are obtained in Table 4.
Error analysis of example: Theoretically, besides the installation error which is determined by operators, another major error source is the interpolation error. Suppose ‫ܣ‬ ‫ݔ(‬ , ‫ݕ‬ ) and ‫ܣ‬ ାଵ ‫ݔ(‬ ାଵ , ‫ݕ‬ ାଵ ) are two adjacent interpolation nodes, ‫‬ , ‫‬ ାଵ , ߮ , ߮ ାଵ are their polar radius and polar angle, respectively. There is: ‫ܣ‬ ‫ݔ(‬ , ‫ݕ‬ ) is one point between ‫ܣ‬ and ‫ܣ‬ ାଵ , ‫ݎ‬ and ߮ are its polar radius and polar angle respectively. ‫‬ can be obtained with putting ߮ into Eq. 20. Hence, in order to ensure the interpolation accuracy, there should be ห‫‬ − ‫ݎ‬ ห < ∆, where, ∆ is the allowable error.

CONCLUSION
This study established the mathematical model of whirling helical groove on a 3-axis linkage machine and the approach to calculate the tool path provides a valuable basis for the machining of helical groove. In addition, the case study including error analysis given in this study also showed that the proposed method is effective in calculating tool path, in terms of machine motion along X, Z and C axes.