Abstract—The helical cutting tools have complex geometries. A rack

cutter is the most economical tool that has been used for manufacturing helical cutting tool. In this paper, the computer program has been designed to evaluate the manufacture abilities following design concept and analyze the technical parameters of helical cutting tool. The program can simulate the sections of helical cutting tool and the rack cutter, analyze the clearance angle, relief angle, width top of the helical cutting tool, and modify the rack cutter profile to show the helical cutting tool profile suitably. This program can predict the differences during manufacture process and offer the best solution for economical consideration.

Index Terms—Computer aided design, helical cutting tool, rack cutter, theory of gearing.

I. INTRODUCTION The development of information technology supports the

process of mechanical manufacture to very high level and makes good profit on mechanical industry. Computer programs have been developed rapidly in the mechanical process, especially for the manufacture of cutters and design of cutters [1], [2]. Helical cutting tools have the important role in the manufacture of machine parts. Rack cutter has been designed for manufacturing helical cutting tool. The personal computer is applied to design the cutter and show the profile of cutters [3]-[5]. Before the manufacture of helical cutting tool, we can simulate the section of helical cutting tool that is cut by rack cutter. Therefore, we can avoid unpredicted errors after manufacturing.

In this study, the computer program has been designed for general purpose, the helical cutter applied for the re-sharpening of pencils is investigated. Some important functions are included in this program. The user inputs parameters, then the program will calculate automatically to show results and analysis. It’s convenient and reliable for the customer.

II. DESIGN OF RACK CUTTER The phenomenon of undercutting has been applied by a

straight-sided hob cutter to generate the profile of the helical cutting tool [4]. Fig.1 shows an example of normal tooth section of hob cutter.

The cutting face can be divided into six regions: (I) the

Manuscript received February 09, 2012; revised March 30, 2012. Ngoc–Thiem Vu, Shinn–Liang Chang are with Department of

Mechanical and Electro-Mechanical Engineering, National Formosa University, 64 WunHua Road, Huwei, 632 Yunlin, Taiwan(a ngocthiem3g@gmail.com (graduate student), b changsl@nfu.edu.tw (professor, corresponding author).

Jackson Hu, and Tacker Wang are with AMAX MFG. CO., LTD.68, Kuang-Cheng Road, TaliCity, Taichung Hsien 41278, Taiwan ( jhu@apexon.com.tw, twang@apexon.com.tw).

left cutting face, (II) the right cutting face, (III and IV) the fillet cutting faces, (V) the top land cutting face, and (VI) the chamfering cutting face. The equations of designed rack profiles of the hob cutter, and the theory of gearing are applied, so the mathematical model of the helical cutting tool can be derived.

In Fig.1, is the origin of the coordinate system , , , it located at the middle of the rack cutter body. The equations of the six regions of the rack cutter in the

coordinate system , , can be obtained but only the equation of left cutting face is shown here as example. The geometrical properties and theory of gearing can be applied to find the equations of other regions.

Fig.1. Normal section tooth profile of hob cutter.

The equation of left cutting face I is presented in the coordinate system as:

=

. sin . cos. tan . cos . sin01 (1)

: Parameter indicates the position on the left cutting face.

III. EQUATION OF THE HELICAL RACK CUTTER The normal section of rack cutter is transferred along the

direction of the lead that is shown in Fig.2. We transform the equations of the cutting face from the rack cutter coordinate system to the helical rack cutter coordinate system, we can obtain the equation of the helical rack cutter.

The transformation matrix M indicates the transformation of the rack cutter coordinate system to the helical coordinate system that is shown in Fig.2 .

Computer –Aided Design of Helical Cutting Tools Ngoc–Thiem Vu, Shinn–Liang Chang, Jackson Hu, and Tacker Wang

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M =

1 0 0 00 sin cos u. cos0 cos sin u. sin0 0 0 1 (2)

The upper sign of M indicates the right-hand helix of

the helical rack cutter and the lower sign of M indicates the left-hand helix.

The equation of left cutting face of the helical rack cutter showed in the coordinate system (region I)

. (3)

Fig. 2. The coordinate system of the right-hand helix of the rack cutter.

Applying the same method for other regions (region II-

VI), the equation of helical rack cutter of 5 regions can be obtained.

IV. EQUATION OF THE HELICAL CUTTING TOOL

Fig. 3. Coordinate system relationship of the rack cutter and

generated gear.

A. Locus Equations Transforming the equation of the cutting face from the

coordinate system of the helical rack cutter to the coordinate system of the helical cutting tool is shown in Fig.3. The transformation matrix M is shown below. The locus equation of the helical cutting tool can be obtained.

M M M (4)

The locus equation of the rack cutter for region I, left cutting face, is shown below:

. (5)

where is shown in equation (3).

Applying the same method for other regions (region II-VI), the locus equation of the full profile can be obtained.

B. Equations of Meshing In Fig. 3, the helical cutting tool is generated by the rack

cutter. Using the theory of gearing, the relative velocity of the contact point ( ) and the unit normal vectors of the helical rack cutter ( ) are obtained. Then, the equation of meshing . =0 can be obtained.

The equation of meshing of the left cutting face is derived as below: . cos . tan . cos

tan 45 2 . tan . sin . tan . (6)

Solving equation (6) and equation (5) simultaneously, the

generated tooth profile by region I can be obtained. Applying the same method for other regions (region II-

VI), the generated tooth profile of the other regions can be obtained.

V. PROGRAM SUPPORTS DESIGNING RACK CUTTER The development of this program can automatically

analyze some technical characteristics and simulate sections of rack cutter and helical cutting tool. The different profiles and optimal design can be predicted. We can estimate the manufacture abilities to save time and money for manufacturers, and enhance the manufacturing efficiency.

The parameters of helical cutting tool and rack cutter can be modified for finding optimal cases. Finally, we can save the modified data in text file or multiple points of section to import into AutoCAD for checking profile again. The computer program is a window application program which works on Window 7 or Window XP using Visual Basic language.

A. Flow Chart of the Program A flow chart of the program for designing the rack cutter

is shown in Fig.4. Input parameters are filled firstly. Then, the sections of helical cutting tool and helical rack cutter can be displayed. If we accept those sections, we can continue for analyzing clearance angle, relief angle, and top land width of the helical cutting tool. The technical parameters of cutters can be checked. Then, we can modify input parameters to show new sections of helical cutting tool and helical rack cutter. Finally, we can choose the best solution and save data for manufacturing.

B. Computer Program The main menu of the program is shown in Fig.5

consisting of File, Edit, and Examples modes, and three tabs. In the tab “Section of the Cutting Tools”, we can input parameters of the helical cutting tool then click on the functions to display the helical cutting tool section or one tooth section. Then, we can evaluate the left cutting face, right cutting face, fillet cutting face, top land cutting face, and chamfering cutting face to choose the compatible rack

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cutter in the next tab named “Technical Analysis Graphs”. The second tab is shown in Fig.6 consisting of displaying

rack cutter profile function, analyzing clearance angle, relief angle, and top land width. In addition, this tab contains special functions such as parameters of helical cutting tools and rack cutter that can be exported and saved in the text files. And, the multiple points in the 2D coordinate of the helical cutting tool section can also selected to save in the other text file.

The third tab “Checking Rack Cutter” is shown in Fig.7, we can modify the parameters to show new section of rack cutter and helical cutting tool. We can evaluate the new sections and compare with the old sections for choosing the best choices for manufacturing.

Fig. 4. Flow chart of the program.

C. Example If the data is inputted as shown in Fig.5, we obtain the

sections and technical parameters of helical cutting tool and rack cutter shown in Fig.5, Fig.6, and Fig.7. In addition, we can use those sections as original sections to compare with modified sections of helical cutting tool and helical rack cutter.

1) Parameters of Helical Cutting Tool Number of teeth, T =12 Outside diameter, D=15.37(mm) Root diameter, d=12.4(mm) Rake angle, α =29 Helical angle, deg =60 Module, m= 1.28

2) Parameters of Rack Cutter Pressure angle of left cutting face, =40 Pressure angle of right cutting face, = 4

Clearance angle (90 Pressure angle of chamfering) = 30 Radius of helical rack cutter, r = 0.15 (mm) Addendum, HKW=1.24 (mm) Dedendum, HFW=0.24 (mm) Tooth thickness of rack cutter, 2 = 5.4 (mm)

Focusing in the third tab in Fig.7, if we want to modify the profile of rack cutter and helical cutting tool, we can change the parameters in each data box. In this example the parameters are modified in Fig.8 shown below: Pressure angle of left cutting face, =45 Pressure angle of right cutting face, = 5 Radius of helical rack cutter, r = 0.4 (mm) Pressure angle of chamfering = 65

Fig. 5. Input parameters and the section of the cutting tools.

Fig. 6. Properties of helical cutting tool and profile of rack cutter.

Fig. 7. The modifying field of HCTA program.

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We can obtain the results in Fig.8, Fig. 9, and Fig. 10.

D. Choosing Improper Parameters Causes the Wrong Result The fail section of helical cutting tool is shown in the

Fig.11 and Fig.12 when the input parameters are improper. Fig.11 shows the failure of section when increase number of teeth of cutter from 12 teeth to 14 teeth. Intersection of left cutting face and right cutting face is on a circle with smaller diameter than standard circle. It can’t be accepted. Fig.12 shows another improper input parameters when we decrease the helical angle of cutter from 60 degrees to 45 degrees. On the other hand, if the other parameters are changed to be unsuitable values, the HCTA program can predict and evaluate the unable ability for manufacturing.

Fig. 8. Modified profile of rack cutter and helical cutting tool.

Fig. 9. Rack cutter is modified and before.

Fig. 10. Helical cutting tool is modified and before.

VI. DISCUSSION When the helical cutting tool is designed, designers can

check the profile of cutter by using HCTA program. When the result is proper as Fig.5, Fig.6, and Fig.7 shown, we accept the input parameters and save them for manufacturing the cutter. In addition, we can modify some

input parameters to get the better profile of the cutter as the mentioned example in the previous section. The new profile of rack cutter and helical cutting tool can be obtained in Fig.8, Fig.9 and Fig.10. Although, the two sets of input parameter both can be accepted. When we input the improper parameters as in Fig. 11, Fig.11 and Fig.12 show the improper profiles of helical cutting tool. The crossing section of the left cutting face and the top land cutting face are intersected on the smaller circle than the required circle.

Fig. 11. Wrong section when entering parameter is improper.

Fig. 12. Wrong section when entering parameter is improper.

VII. CONCLUSION In this study, the computer program has been designed to

simulate and modify the sections of helical cutting tool and helical rack cutter. Before we manufacture the cutters, we can simulate the profiles of cutters using this program to display the sections and technical characteristics. Then, we evaluate the producing abilities and predict the differences of cutters after manufacturing. This program is written by Visual Basic language with simple interface helping users use easily.

This program design not only supporting for manufacture but also helping learners to study this field easily.

APPENDIX 2 Tooth thickness of the rack cutter C Shifted amount D Outside diameter of the helical

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cutting tool e Height of chamfering from point q to the

root of tooth of the rack cutter HKW :Addendum of the rack cutter HFW: Dedendum of the rack cutter , , : Parameter of vector , , respectively

Origin of coordinate system Circular pitch of the rack cutter

r Outside radius of the helical cutting tool Radius of pitch circle of the helical

cutting tool , ,μ: Parameters of , , R Radius of the rack cutter fillet

Fixed coordinate system Coordinate system of helical rack cutter Coordinate system of helical cutting tool

u Distance between origins and λ Lead angle of the helical cutting tool

Angular displacement of the helical cutting tool while hobbing , , : Pressure angle of cutting edge I, II, and VI respectively.

ACKNOWLEDGMENT The work outline in this paper was supported by APEX

MFG. CO., LTD and the National Science Council under grants NSC91-2212-E-150-022 and NSC92-2212-E-150-033.

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[2] J. D. Kim and D. S. Kim, “The development of software for shaving cutter design,” Journal of materials processing technology, vol. 59, pp. 359-366, 1996.

[3] F. L. Litvin, “Gear geometry and applied theory, second edition,” Published by Cambridge University press, September 2004.

[4] S. L. Chang and H. C. Tseng, “Design of a novel cutter for manufacturing helical cutting tools,” Proceeding of the institution of mechanical engineers, Journal of Mechanical Engineering Science vol. 219, pp.395-408, 2005.

[5] J. K. Hsieh, H. C. Tseng, and S. L.Chang, “Novel hob cutter design for the manufacture of spur-typed cutters,” Journal of materials processing technology, vol. 209, pp. 847-855, 2009.

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