Spur Gear Design With An S-Shaped Transition Curve Application Using MATHEMATICA And CAD Tools

S. H. Yahaya1, M. S. Salleh3 Faculty of Manufacturing Engineering Universiti Teknikal Malaysia Melaka

Melaka, Malaysia 1saifudin@utem.edu.my, 3shukor@utem.edu.my

J. M. Ali2 School of Mathematical Sciences

Universiti Sains Malaysia Penang, Malaysia

2jamaluma@cs.usm.my

Abstract—The aim of this paper is to introduce the application of an S-shaped transition curve in designing and generating a spur gear. In engineering field, the spur gear usually designed by using an involute curve where this curve needs to use a tracing point method to obtain a curve while an S-shaped curve can obtain a curve directly. The comparative study between an S-shaped method and direct method in Computer Aided Design (CAD) tool also covered.

Keywords-S-shaped transition curve; Curvature continuity; CAD tool; Mathematica; Spur gear

I. INTRODUCTION Curve or surface design is very vital in Computer Aided

Design (CAD) or in Computer Aided Geometric Design (CAGD). In this area, curve or surface is designed by using a certain function with several curve or surface properties. One of the examples in curve design is transition curve. The transition curve is defined as a curve that has a specific radius and degree of curvature which means that the curvature changes either increasing or decreasing along the curve length. Transition curve also can be called as an easement. According to Baass [1], five transition curve cases in highway design have been identified, one of the cases is circle to circle with an S transition curve. This design template will be used and applied in this paper.

The function that used to design an S-transition curve is Bézier-like cubic which this function in Bézier form. The function has the lowest degree of polynomial and also has a shape parameter where this parameter used to control the curve shape. The Bézier form is usually used in CAD and CAGD application because of its geometric, numerical properties and easy to implement. The main objective of this paper is to design a spur gear by using an S-shaped transition curve and design template from Baass [1]. In manufacturing engineering, designer or manufacturer usually used an involute curve where the application of tracing point method is applied to obtain the curve while an S-transition curve can directly create the curve.

Many related papers have been produced in this area. This includes planar G2 transition between two circles with a fair cubic Bézier curve and a planar cubic Bézier spiral written by Walton and Meek [2, 3], Habib and Sakai [4–7] have summarized G2 cubic transition between two circles with shape control, G2 planar spiral cubic interpolation to a

spiral, G2 planar cubic transition between two circles and circle to circle transition with a single cubic spiral. Sheveleva et al. [8] have proposed algorithms for analysis of meshing and contact of spiral bevel gear while the computer aided design of elliptical gear has studied by Bair [9]. Hwang and Hsieh [10] have discussed about the determination of surface singularities of a cycloidal gear drive with inner meshing.

The proposition of this paper begins with some notation and convection which are needed in preparing this paper. The next discussion will touch on the introduction of Bézier-like cubic function, explanation on circle to circle with an S-shaped transition curve together with some numerical examples. After that, the discussion on spur gear tooth design using an S-shaped transition curve application and coordinate selection using Mathematica 6.0 will be covered. Finally, spur gear solid will be designed by using CATIA V5 as a CAD tool from the selected coordinates with some conclusion and several recommendations will be included for future work.

II. NOTATION AND CONVECTION Consider the Cartesian coordinate system such as vector,

).,( yAxA=A The Euclidean norm or length of vector A is

denoted by ,2y

2x

AA +=A with an angle measured in this

paper is anti-clockwise angle. The derivative of a function f is denoted by .f ′ The dot product of two vectors, A and

B is written as A·B. A planar parametric curve is defined by a set of points,

))(),(()( tytxtz = with t given in real line interval. In this paper, we use t ].1,0[∈ The cross product of two vectors, A and B is symbolized by ∧ referring to Juhász [11], which is defined by A^B = sinθxyyx BABABA =− where θ is anti-clockwise angle. The tangent vector of a plane parametric curve is stated by .)(tz ′ If 0)( ≠′ tz then the definition of curvature )(tz can be defined as,

3)(

)()()(

tz

tztzt

′

′′∧′=κ (1)

2009 International Conference on Computer Technology and Development

978-0-7695-3892-1/09 $26.00 © 2009 IEEE

DOI 10.1109/ICCTD.2009.76

426

2009 International Conference on Computer Technology and Development

978-0-7695-3892-1/09 $26.00 © 2009 IEEE

DOI 10.1109/ICCTD.2009.76

426

The derivative of Equation (1) will express as,

5)(

)()(

tz

tt

′=′

ωκ

(2) where,

)()(

)()(3)()(2

)()(

tztz

tztztztzdt

dtzt

′′⋅′

′′∧′−′′∧′′=ω

The next section will explain about Bézier-like cubic

function with the shape parameters.

III. BÉZIER-LIKE CUBIC FUNCTION Consider a Bézier-like cubic function taken from Ali et

al. [12] as shown below:

]1,0[,3))12)(1(1(2(2)2)1(1(

1))2)1(0(0)))02(1(2)1(()(

∈−−++−

+−+−+−=

tPttPtt

PttPtttz

λλ

λλ

(3) where 3,2,1,0 PPPP are the control points and 1,0 λλ are the parameters controlling the curve shape. In this paper, the value of )3,0(1,0 ∈λλ to guarantee the Bézier- like cubic function does not change the sign or is a positive. The next discussion will focus on circle to circle with an S-shaped transition curve using Bézier-like cubic function.

IV. CIRCLE TO CIRCLE WITH AN S-SHAPED TRANSITION CURVE

By using the Equation (3) with the control points as defined by Habib and Sakai [6] with some modifications are made where,

)*(),*(

),*(),*(

3113232

1010000trcPtkPP

thPPtrcP

+=+=

−=−=

(4) with the knot points are,

]sin[],cos[3

],cos[],sin[2

]cos[],sin[1],sin[],cos[0ββββ

αααα

=−=

−==

tt

tt

(5) with 1,0 cc are the centers of two circles, Ω0, Ω1, 1,0 rr are

the radii of two circles, Ω0, Ω1, 0,1 PPh = , 2,3 PPk = and

βα, are the angles of controlling two circles, Ω0, Ω1. An S-shaped transition curve is designed by using curvature continuity where this continuity has a standard condition which shown below.

1

11

0

10

rt

rt ==−== )(,)( κκ

(6)

In order to design an S-shaped transition curve, these two conditions in Equation (6) must be fulfilled. Equation (6) will also be used to estimate the parameter values, h and .k A few examples will be discussed in the next section to ensure the theoretical part above is correct.

Figure 1. Circle to circle with an S-shaped transition curve.

V. NUMERICAL EXAMPLES

1. Let we have ,,, 1 111 10 =−= cc and .110 == rr The α and β values used are rad 26892. and rad 18172. with

the shape parameters, 10 λλ , equal to ..52 We find that the parameters kh, equivalent to 90. after applied all the equations above. This example is shown in Figure 2.

Figure 2. Example 1.

2. Let assume ,,,. 1 111 500 −== cc with 500 .=r and

.11 =r The angles employed are rad 22171.=α and

rad 83973.=β with 810 .=λ and .21 =λ We discover that h and k approximate to 70. and ..90 The result is demonstrated in Figure 3.

Figure 3. Example 2.

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From the examples above, all the conditions in Equation (6) have been fulfilled. The next discussion will be the explanation of designing a spur gear using an S-shaped transition curve where the algorithm above is applied.

VI. SPUR GEAR TOOTH DESIGN The curve most commonly used for gear tooth profiles is

involute curve. The transition curve application is popular used in highway or railway whereas in this paper, we would like to design the spur gear tooth profile by applying an S-transition curve.

Let we design spur gear profile with six tooth set. We employ center, c0 , 00= with r0 1= and r1 ..5880= Inside these two radii, we have another radii, r equivalents to

..2060 The shape parameters, 10 λλ , used are 52. and 81. with angles in ].,.[ radrad 23605 52360 based on the quadrant. By applying all the inputs, we acquire that the parameters, kh, approximate to 30. and ..40 The spur gear profile is visualized by using Mathematica 6.0 software and the result is shown in Figure 4.

Figure 4. Spur gear tooth profile using an S-shaped transition curve.

After the gear profile is completed, the next process is to build up the 3D solid model. The model will be designed by using CATIA V5 software. This software is used because its capability in designing 3D solid model. In addition, the gear tooth shape that will be designed by using CATIA V5 must be the same as in Figure 4 because an S-shaped curve has been used to produce the tooth as in Figure 4. In order to do that, the coordinate will be selected along the generated curve (Figure 4) by using Mathematica 6.0 software. After that, the selected coordinate will be used to design a solid model. The detail of these procedures will be discussed in the next section.

VII. COORDINATE SELECTION In Mathematica 6.0, there are several drawing tools in

graphics palette. One of the tools is “get coordinates” as shown in Figure 5.

Figure 5. Drawing tools in Mathematica 6.0.

We click the “get coordinates” tool and move the mouse pointer over the 2D graphics or 2D plot. The approximate coordinate values of mouse pointer are displayed. Then, click to mark the coordinates. We can click at other position to add markers. The sample is presented in Figure 6.

Figure 6. Example using “get coordinates” tool.

Finally, use Ctrl+C to copy the marked coordinates and Ctrl+V to paste these coordinates into an input cell as shown in Figure 7.

Figure 7. List of coordinates in an input cell.

VIII. SPUR GEAR SOLID In designing a spur gear model, the procedure should

start by graphing 2D spur gear using CATIA V5. We redesign the spur gear from the selected coordinates. In this paper, 34 coordinates are chosen and the spline package in CATIA V5 will be applied to join all the coordinates. All the fundamental aspects in creating a spur gear also included such as an outer circle and the root circle. We also made some adjustments in this design where the gear thickness is set to be 4 mm, gear shaft radii equivalents to 2.836 mm and outer circle radii is 20.02 mm. The adjustment is very much needed especially in gear shaft to give more strength to the model and visually pleasant (Figure 8). However, the tooth shape remains the same as in Figure 4.

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Figure 8. Spur gear tooth profile and its solid model from an S-shaped method using CATIA V5.

In this paper, we also show that the spur gear solid as designed by using a direct CATIA method (a method using an involute curve) is used to design this spur solid. All the packages in CATIA V5 are applied except for spline package. Normally, spline package is used for the coordinates or points. All the measurements above such as gear shaft radii are maintained. The design of this solid is illustrated in Figure 9.

Figure 9. Spur gear solid using direct CATIA method.

According to both methods, an S-shaped method, which is more on mathematical features such as a certain formulae, must be defined or the design parameters must be declared. The direct method reflects to an engineering drawing style.

IX. CONCLUSION AND RECOMMENDATIONS This research shows that an S-shaped transition curve can

be applied in gear tooth design particularly for spur gear. The tooth shapes from the findings almost the same. It can be seen clearly in Figure 4, Figure 8 and Figure 9. The integration of softwares Mathematica 6.0 and CATIA V5 is very useful in this research. In future, gear design analysis will be carried out to know the applicability of the design.

The analysis covers about the dynamic analysis and gear noise modeling. Once all the analyses have been done, this spur model will be fabricated by using Computer Numerical Control (CNC).

ACKNOWLEDGMENT This research was supported by UTeM under research

FRGS grant, FRGS/2007/FKP (17) – F0042. The authors gratefully acknowledge to everybody for their helpful comments.

REFERENCES [1] K. G. Baass, “The use of the clothoid templates in highway design,”

Transportation Forum 1, 1984, pp. 47–52. [2] D. J. Walton and D. S. Meek, “Planar G2 transition between two

circles with a fair cubic Bézier curve,” Computer Aided Design, 31, 1999, pp. 857–866.

[3] D. J. Walton and D. S. Meek, “A planar cubic Bézier spiral,” Journal of Computational and Applied Mathematics, 72, 1996, pp. 85–100.

[4] Z. Habib and M. Sakai, “G2 cubic transition between two circles with shape control,” Journal of Computational and Applied Mathematics, 2008.

[5] Z. Habib and M. Sakai, “G2 planar spiral cubic interpolation to a spiral,” Proceedings of the Sixth International Conference on Information Visualisation, 2002.

[6] Z. Habib and M. Sakai, “G2 planar cubic transition between two circles,” International Journal of Computer Mathematics, vol. 8, 2003, pp. 959–967.

[7] Z. Habib and M. Sakai, “Circle to circle transition with an a single cubic spiral,” Proceedings of the Fifth IASTED International Conference Visualization, Imaging, and Image Processing, 2005, pp. 691–696.

[8] G. I. Sheveleva, A. E. Volkov, and V. I. Medvedev, “Algorithm for analysis of meshing and contact of spiral bevel gears,” Mechanism and Machine Theory, vol. 42, 2006, pp. 198–215.

[9] B. W. Bair, “Computer aided design of elliptical gears,” Mechanical Design, vol. 124, 2002, pp. 787–793.

[10] Y. W. Hwang and C. F. Hsieh, “Determination of surface singularities of a cycloidal gear drive with inner meshing,” Mathematical and Computer Modelling, 45, 2007, pp. 430–354.

[11] I. Juhász, “Cubic parametric curves of given tangent and curvature,” Computer-Aided Design, 25(1), 1998, pp. 1–9.

[12] J. M. Ali, H. B. Said and A. A. Majid, “Shape control of parametric cubic curves,” Proceedings of the Fourth International Conference on CAD/CG, 1995, pp. 161–166.

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