Design& Analysis of Precession Polygon hole drilling Tool

Design& Analysis of Precession Polygon hole drilling Tool

Shaik Shajahana*

, V.Diwakar Reddy b,M.Venu Gopal Naidu

c

aDepartment of Mechanical Engineering, Sri Venkateswara University College of Engineering, Tirupati, A.P, India bProfessor, Department of Mechanical Engineering, Sri Venkateswara University College of Engineering, Tirupati, A.P, India

cResearch Advisor, Department of Mechanical Engineering, Sri Venkateswara University College of Engineering, Tirupati, A.P, India

Abstract

Drilling is the operation of making a hole by removing a volume of metal from the job by a

rotating cutting tool. It is one of the most versatile process to cut holes on the workpiece. Drilling is usually

performed to obtain circular holes. But it can be used to produce polygon shape holes. A polygon hole can be

produced by many procedures in which broaching process is a prominent one. The other techniques include wire-

Electro Discharge Machining (Wire-EDM), Ultra Sonic Machining (USM), Water Jet Machining (WJM), etc.

which increases the cost of manufacturing and complexity in obtaining blind holes. In this paper,the drilling of

polygon holes by a special tool called reuleaux polygon is discussed. Reuleaux polygon is a special geometry in

which the tool has (n-1) sided cutting edges for an (n) sided polygon hole. The regular rotating motion is used to

cut the polygon hole by converting this rotatory motion into a polygon path tracing motion by using a floating

tool. The floating tool is coupled to the rotating chuck using Oldham coupling. One end of the cutting tool rotates

inside the polygon guide while the driving end will be attached to the standard drill press along with other small

attachments. A mathematical relation is derived to correlate the geometry of cutting tool in accordance with the

geometry of the required polygon hole. The best possible process is chosen for the required applications through

this paper.

Keywords: Reuleaux Triangle; Reuleaux Polygon; Oldham Coupling.

1. Introduction

The problem of drilling non-circular holes is one of long standing engineering interest. In 1914, James

Watts came up with an idea of rotating a reuleaux triangle within a square which resulted in tracing a square

by the corners of the reuleaux triangle. A tool was designed with the reuleaux triangle by sharpening its

edges. The three cutting bits can trace out a curve which is almost square.

The Reuleaux Triangle is one example of a wide class of geometrical discoveries like the Mobius strip

that did not find many practical applications until relatively late in humankind’s intellectual development.

There exist a number of machines like drilling, milling machines in the market to produce circular holes. But

for polygon or any other type of holes, the methods like Broaching, Wire-EDM, and Ultra Sonic Machining

can be used. These processes are very expensive and require special tools for machining.

2. Literature Review

A reuleaux polygon is a shape generated by a number of arcs joining the vertices of the polygon centred

at the opposite vertices. Reuleaux polygon has a special property that it can rotate completely between two

parallel supporting lines regardless of their orientations. In any pair of parallel lines, one of the two lines will

necessarily touch the polygon at its vertex [1]. The other supporting line may touch the polygon on the

opposite arc and their distance equals to the radius of the arc. The Reuleaux triangle is the first of a sequence

of Reuleaux polygons, whose boundaries are curves of constant width formed from regular polygons with an

odd number of sides. Some of these curves have been used as the shapes of coins.

An attempt has been made by Michael Goldberg in his paper to identify the surfaces inside which the

rotors can be made to rotate by forming the outer surfaces to be convex and there by identifying

mathematically the trace produced by the points on the rotor. The rotor could not trace the required

polygon[5].

A method is formulatedby Barry cox and Stan wagon to design tool to obtain square and hexagonal

holes. But it can be achieved only with a single point cutting tool. From this, it is established that it can trace

a square or hexagon with a single point [6].There is another process to design the tool bits that can produce

odd-sided polygons. This method of designing is based on a rule that the envelope formed by a vertex of a

Science, Technology and Development

Volume X Issue XI NOVEMBER 2021

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triangle under the constraint that the other two sides maintain tangential contact with two fixed congruent

circles is a circular arc [7].

There have been many other papers explaining the design of the tool bit for obtaining a square but these

two papers remained base for the new technological innovations.

3. Development of Reuleaux Polygon

3.1 Construction of Reuleaux Polygon

The shape of the ‘n’ sided reuleaux polygon is generally obtained from a set of ‘n’ circles overlapped in a

regular pattern forming a closure as shown in the figure 1. But this process needs complex positioning of circles in

order to be equidistant from the others. Hence the reuleaux polygon is drawn from the regular polygon itself. The

sides of the regular polygon are replaced with curves centered at the opposite vertex.

(a) Pentagon (b) Heptagon

Fig. 1: Construction of Reuleaux Polygons

In the above figure, it can be observed that the reuleaux polygon rotor shape is formed by arranging the

number of circles which is equal to the number of sides of the polygon rotor in a circular pattern. All the circles

would be equidistant from the circles beside them. The overlap of the circles gives a shape which is the required

reuleaux polygon or rotor. This is a tedious process in which the arrangement of the circles would be a hard task.

There is an alternate way of obtaining the reuleaux rotors.

The second method of designing reuleaux rotors is by starting with the polygon itself. The sides of the

polygon are replaced with curves which are drawn from the opposite vertex.

(a) Pentagon (b) Heptagon.

Fig. 2Final shapes of rotors



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The reuleaux polygons have the property of maintaining equal distance between two parallel lines which

gives them the ability to trace the polygons without slipping against them.

3.2Mathematical formulation

The rule for drilling a regular ‘n’ sided polygon where n is even,

Length of the longest diagonal of a ‘n-1’ polygon (tool)

= twice the length of the apothem of a ‘n’ sided Polygon (hole)

Length of the longest diagonal of an ‘n’ sided polygon,

� � �2 sin � ∵ ℎ��

� � �sin �� ∵ ℎ��

Length of apothem of an ‘n’ sided polygon,

� � �2 tan ��

Fig. 3 Odd sided reuleaux polygon inside an even sided regular polygon

If the has to be ‘n’ sided, the tool should be ‘n-1’ sided.

Hence by the rule,

Dn-1 = 2An

Let, the side of polygon tool is taken as ‘t’ and the side of polygon hole is taken as ‘h’,

------------ Eq 1

------------ Eq 2

------------ Eq 3

------------ Eq 4

------------ Eq 5



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�2 sin �� 2 � ℎ2 tan ��

�

Hence the side of the polygon of tool can be given by,

� � �2 sin ��tan �� ℎ

This formula (Eq 6) provides the relationship between the required sizes of tool to drill an even sided

polygon hole. But it comes at the cost of some errors like rounded corners which need to be machined further to

make it perfect polygon as shown in figure 4. The shape obtained is represented in dotted lines.

(a)Pentagon (b) Heptagon.

Fig. 4Shapes obtained in this process

3.3Error Optimization

Error(%) � � !" #$ %! $!&' ()"*! �� !" #$ ()"*! #+'",!- , '),. * #&!..� !" #$ %! $!&' ()"*! / 100

The percentage of error arising with each polygon is obtained as mentioned in the below table:

Table 1 Error Optimization

Shape Error (%)

Square

Hexagon

Octagon

Decagon

1.17

0.98

0.71

0.52

The errors are calculated based on the sizes of the holes as length of side = 50mm.

From the table 1 it can be observed that the error is getting reduced with increase in number of

sides of the regular polygon. The above errors are for even sided polygon holes. Theoretically,

the above process of drilling odd sided holes would not create any error.

The error has no change according to the size of the required hole. The above mentioned data is the

maximum limit of the exactness one can achieve in this method. The error can be more than this depending on the

tool design and its cutting efficiency.

------------ Eq 6

------------ Eq 7



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4.Design & Analysis

4.1Design & Assembly

The shape of the tool is designed in such a way to provide maximum removal of chip without hampering

the geometry of the tool. The taper in the inner side of the tool allows drive away the chips formed during the

drilling operation.The main edges of the cutting tool is formed from the reuleaux polygon that traces inside the

guide. It is designed to holdthe maximum fatigue strength. Hence the inner side is bent into the shape of curves

for the best results.

Fig. 5Tool assembly for Hexagonal holes.

In this paper the tracing is done for three shapes i.e. Hexagon, Octagon and Decagon.The shapes of the

cutting tool and its corresponding trace can be observed in the below figures.

(a) For hexagon (b)For Octagon (c) For Decagon

Fig. 6Tool shapes for different polygons



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(a)Hexagon (b) Octagon (c) Decagon

Fig. 7Paths traced by the tools

The hexagon is obtained from the five sided reuleaux tool. The corners of the reuleaux pentagon is

carved into the shape of a cutting tool through which the cutting can take place. The maximum area of the

hexagon can be obtained through this method is 99.01% of the perfect hexagon. The octagon is obtained from the

seven sided reuleaux tool. The maximum area of the octagon can be obtained through this method is 99.29% of

the perfect octagon. The decagon is obtained from the nine sided reuleaux tool. The maximum area of the decagon

can be obtained through this method is 99.48% of the perfect decagon.

4.2Case Study

Problem: Calculate the size of tool bit required to obtain a 6mm sided hexagon.

Solution:

Let us take design a rotor for hexagonal hole,

Side of the hexagonal (n=6) hole required, h = 6mm

The shape of the polygon would be formed from a (n-1) sided polygon.

Hence, the shape of the rotor would be obtained from a pentagon.

The side of the pentagon required would be,

� � �2 sin �2tan ��3� 6

� � 6.4 77

From the 6.4mm sided pentagon, by drawing arcs along the sides from the opposite vertex a reuleaux pentagon is

obtained which can drill a 6mm sided hexagon hole.

The reuleaux pentagon formed is cut to make sharp edges out of the corners.

The edge of the tool bit is made to be 450 for the maximum removal of the material to be machined. The

arc formed during the edge design is prolonged to meet the side of the reuleaux polygon. This process is repeated

along the remaining sides to complete the shape of the rotor tool.

The area of the required hexagon with a side 6mm would be AH = 93.53 mm2.

The area of the hexagon that can be obtained from the above designed rotor would be A = 92.3 mm2.

The error would be around 1.31% which can be rectified by the finishing processes.

The main cutting edges of the tool is formed from the reuleaux polygon that traces inside the guide. The

edges are made in such a way so as to have maximum fatigue strength. Hence the inner side is bent into the shape

of curves for the best results.

4.3 Analysis on reuleaux tool Assembly

The reuleaux tool designed in this paper is made to run in explicit dynamic analysis in Ansys to obtain

the feasibility of tool in the working conditions. The tool material is Titanium and the work piece material

isAluminium 1100-O. It is found out that the tool design can be used in drilling the polygon shaped holes.The

results of the analysis are obtained as the following.

The tool is meshed into 11615 elements and 7164 nodes. The input load in terms of rotational velocity

500 rad/sec is applied on the tool assembly along with a downward motion (-z).



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4.3.1Total Deformation

Fig. 8

4.3.2 Equivalent Elastic Strain

Fig. 9

4.3.3 Equivalent Stress

5.Analysis Results

The analysis of tool in explicit dynamics of A

Titanium and the work material is taken to be

is given an upward displacement. With the given conditions, the Total deformation, Eq

Equivalent stress are obtained from the analysis. The result obtained is from a short span of analysis but the

software included the output which will happen in longer durations.

Fig. 8Total Deformation and its plot

Fig. 9Equivalent elastic strain and its plot

Fig. 10Equivalent stress and its plot

f tool in explicit dynamics of Ansys produced the below result. The tool is taken to be of

Titanium and the work material is taken to be Aluminium. The tool is given a rotatory motion and the work piece

upward displacement. With the given conditions, the Total deformation, Equivalent elastic strain and


software included the output which will happen in longer durations.

nsys produced the below result. The tool is taken to be of

. The tool is given a rotatory motion and the work piece

uivalent elastic strain and




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Table 2Analysis Results for the rotational velocity of 500 rad/sec.

Results

Total Deformation

(mm)

Equivalent Elastic Strain

(mm/mm)

Equivalent (Von-Mises)

Stress (MPa)

Minimum 0 0 0

Maximum 11.12 0.46 1009.8

MinimumOccursOn Guide

MaximumOccursOn Tool

MinimumValueOver time

Minimum 0 0 0

Maximum 0 0 0

MaximumValueOverTime

Minimum 1.0695e-014 0 0

Maximum 36.12 0.64 1188

There is another process by which the perfect polygon holes can be obtained. Barry Cox and Stan Wagon

worked on creating a procedure through which perfect polygons can be produced. This idea is fine in creating odd

sided polygons but for even sided polygons, a common formula could not be created for all even sided polygons.

But the problem in these tools is that it works with single point cutting tool. The single point cutting tool traces

the polygon as a marker but not as a drilling mechanism. In this paper, the common formula is generated for the

even sided polygons.

Fig. 11Top view of tool designed by Barry cox and Stan wagon for hexagonal hole.

6. Conclusion

The drilling of polygonal holes in the procedure mentioned by Barry cox and Stan wagon is applicable to

produce odd sided polygonal holes. But it is not applicable for drilling the even sided polygons. The tool is single

pointed and it is not balanced during the process of cutting.In this paper, a common procedure is provided to drill

even sided polygons. The drilling of the polygon holes is done by a balanced mechanism. The error optimization

is carried out for four shapes which are square, hexagon, octagon and decagon. The errors obtained in this process

can be reduced by the finishing processes like grinding. A tool is designed in NXCAD and analysed it in Ansys

Explicit dynamics. The external conditions are provided for the tool is made up of titanium to drill a hole on

aluminium workpiece. Hence the polygon holes can be drilled in this process by minimizing the errors.

References

[1] Franz Reuleaux, Paths of points of the curves-triangle relatively to the square unit-3, Theory of

Kinematics, 1875.

[2] Michel Goldberg, Circular-arc rotors in regular polygons, American Mathematical Monthly 55 pp393-

402, 1948.



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[3] Michel Goldberg, Rotors in Spherical polygons, Journal of Mathematics and Physics, v.30, 1953.

[4] Michel Goldberg, Basic Rotors in Spherical polygons, Journal of Mathematics and Physics, v.34, 1956.

[5] Michel Goldberg, Trammel Rotors in regular polygons, American Mathematical Monthly, v.64, 1957.

[6] Barry Cox & Stan Wagon, Circle-squaring: A mechanical View, College Mathematics Journal, 40 238–

247,2009.

[7] Barry Cox & Stan Wagon, Drilling For Polygon, The American Mathematical monthly, 119:4, 300-312,

DOI: 10.4169/amer.math.monthly.119.04.300., 2012.

[8] Shailesh S. Sengar, VaibhavRaghav and ChadaramSrinivasu, Design & Fabrication of a Special Tool to

Produce Square Hole, 3rd International Conference on Materials Processing and Characterisation

(ICMPC), 2014.

[9] Ridha Alwan Ahmed, Mathematical Analysis of Square hole Drilling Mechanism, Journal of

Engineering and Sustinable Development, 2018.

[10] Reuleaux polygons from Wolfram math - https://mathworld.wolfram.com/ReuleauxPolygon.html.



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Documents

Design& Analysis of Precession Polygon hole drilling Tool