Boolean Operation Of Coupler Curve With Simple Polygon

This document contains :

• Introduction
• Applications
• Algorithm
• Sample Results
• References

Introduction

Our project can be best introduced with the following two questions:

• What is meant by a Coupler Curve ?
• What is the need of doing Boolean Operations of the Coupler Curve with a polygon ?

All we are familiar with four bar Mechanisms. These can be divided into two broad catageries:

1. Crank-lever Mechanism: In this class, one of the arm makes full rotation and the other makes only oscillations.
2. Double lever Mechanisms: In this class, both the members are oscillating @ their pivots

The member joining the crank and the follower is known as coupler. And the path generated by any point of the coupler is known as Coupler Curve. Our project is limited to only first class of problems as it is widely used.

Usually a mechanism is designed for the doing the prescribed work. So it is essential to know wheather it is being utilised properly. Also it is important to see wheather it is fouling with any member in the system.

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Applications

Boolean operation of the coupler with the polygon can be utilised for such applications. Following are some of the practical areas which demands the necessity of our projct.

1. To know the span of the motion of the Coupler.
2. To know the intersection of the Coupler Curve with the object in the given space. i.e. % of the motion being utilised for the actual work.
3. To know the % of the unutilised motion of the Coupler.
4. To know the % of the object having no interaction with the Coupler.etc.

Following are some of the applications in day-today's industrial life.

1. When a cutting tool is mounted on the Coupler we need to know the volume of the metal removed and the balanced material left.
2. Mechanisms are popularly used in Earth Moving Equipments. On the same line we would like to know the stock transferred.
3. A spot welding gun is mounted on the Coupler and welding of a car body shell is being done. Intersection of the Curve and the shell will give the area covered. Also inter. with the fixtures will indicate the foulling of the gun with it.

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Algorithm

• Input to the Code:

1. Coordinated motion of the crank and follower.(3 positions)
2. Length and angle subtended by the Coupler.
3. Coordinates of the polygon.
4. Choice of the Boolean operation.

Output of the Code:

1. % Effective utilisation.
2. % Unutilisation.
3. % Stock left.

Format of the input:

1. Three position of the Input & Output Angles
2. Angle between coupler and the arm.
3. Arm Length.
4. Choice of Boolean Operation.
5. No of vertices of polygon.
6. Co-ordinates of vertices of the polygon.

Steps of the Algorithm:

1. Links are assumed as vectors. Equilibrium Equations in X and Y Directions gives the relation between the link lengths in ters of input and output angles. This is known as FRUDENSTIEN' EQUATION.
2. Given positions are put in the above EQn. Then the three EQns are solved by GAUSS'S ELIMINATION to determine the link length proportions.
3. One of the link length (Fixed distance between the pivots) is assumed and other link lengths are determined from Freudenstien Equation.
4. By comparison of the link lengths, the class of the Mechanism is found out. If Double lever Mech. then the code terminates else continued.
5. Origin is asumed at the center of rotation of the crank. Second point of the crank is calculated by sin-cos values of the radius. Also position of the pivot of the follower is the fixed length along the positive "X" axis is known.
6. Now a circle having Coupler link radius is assumed at the 2nd point of the crank. Another circle of follower length is assumed at another pivot.
7. Intersection of these two circles give the position of the 2nd point of the follower. Positive root is taken for same rotation of the crank and follower.
8. From the given Geometry of the Coupler, third point of the coupler is calculated. i.e. Its orientation w.r.t. crank and follower is determined.
9. Same procedure is repeated for the complete rotation of the crank.
10. Thus locus of all such points of will give a closed curve which is nothing but the COUPLER CURVE.
11. Then the Booean operation is carried out with the given polygon.
12. With the help of TRI-TEST, area of the intersection is determined.

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Sample Results

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References

1. Principle of interactive computer graphics.

   Williamm Newman, Robert Sproul.

2. Graphical display for Engg.documentation.

   Daniel Ryan.

3. Applied linkage synthesis.

   D.C.Tao.

4. Applied kinematics.

   Kurt Hain.

5. Kinematic synthesis of linkages.

   Allan Hall.

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   This HTML Version is prepared by Santosh Kulkarni and Anil Suryawanshi, M.Tech. (I sem-SMD) as per term project of COMPUTER AIDED DESIGN.

   Instructer: Dr. Amitabh Mukerjee.

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