Trammel of Archimedes

 Trammel of Archimedes

Authors : Shreyash Gurav, Parth Gurjar, Sumedh Habde, Adarsh Halake, Harsh Rikame

Points covered in this blog :

• What is Trammel of Archimedes

• Construction of Trammel of Archimedes

• Mathematics

• Applications

• Conclusion

Introduction 

A trammel of Archimedes is a mechanism that produces an ellipse's shape. It comprises two shuttles that are restrained (or "trammeled") in parallel channels or rails and a rod that is connected to the shuttles at fixed points along the rod by pivots.

All locations on the rod travel in elliptical trajectories as the shuttles alternately back and forth along their individual channels. The rod's movement is known as elliptical motion. The lengths of the semi-axes a and b of the ellipses are equal to the separations between the point on the rod and each of the two pivots.

Source : Wikipedia

The pivots' straight lines are particular examples of an ellipse, where one axis' length is equal to two times the distance between them and the other is zero. Such straight lines reciprocate at all locations on a circle whose diameter is defined by the two pivots. The smaller circle in a Tusi pair corresponds to this circle.

A circle revolves around the location where the channels converge at the point located halfway between the pivots. This circle is a particular instance of an ellipse as well. The axes are the same length here. The circle's diameter is determined by the separation between its pivots. The trammel rotates in the opposite direction as it moves around the orbit. This means that if a crank is used to engage the trammel at its halfway point and drive it, the rotation of the trammel and the crankpin will be equal and opposite, which will increase friction and speed up wear in real-world situations. Due to the crank's small throw, which is only 1/4 the travel of the pivots, this is made worse by strong forces.

An ellipsograph is an Archimedes trammel designed to draw, cut, or manufacture ellipses, for instance in sheet materials like wood. The correct tool (pencil, knife, router, etc.) is attached to the rod in an ellipsograph. The ellipse's size and form can typically be adjusted by adjusting the distances a and b.

Under the names Kentucky do-nothings, nothing grinders, do nothing machines, smoke grinders, or bullshit grinders, wooden replicas of Archimedes' trammel have also been made and sold as toys or novelty items. The sliding shuttles' positions are typically set, and the drafting tool is replaced in these toys with a crank handle.

Construction

Source : IRJET Paper

It is made up of two shuttles that are restrained in two parallel channels, which are connected to the frame by pivots placed at certain points along the rod. A slider-I is represented by link 1, a connecting rod by link 2, a slider-II by link 3, and a slider-III by link 3. One slider moves in a horizontal direction, while the other moves in a vertical direction.  Our focus is to get the required approximate straight motion for bottle shaking with the help of an elliptical trammel mechanism.

Mathematics 

Source : Wikipedia

Source : Wikipedia

Let A and B serve as the sliders' pivots, and C serves as the rod's outer end. Let AB and BC represent the separations between points A and B, respectively. Assume that A and B are two sliders that move in accordance with the y and x coordinate axes, respectively. The coordinates of point C are given by when the rod forms an angle with the x-axis.

These are in the form of the standard parametric equations for an ellipse in a canonical position. The further equation

is immediate as well.

The trammel of Archimedes is a specific case of the more generic oblique trammel and is an illustration of a four-bar connection with two sliders and two pivots. The points A, B, and C can form a triangle, and the axes limiting the pivots don't have to be perpendicular. The locus of C that results is still an ellipse.

Applications 

1. It serves as a mechanism for an automatic tool changer.

2. Elliptic shapes are drawn using an elliptical trammel.

They may be used to create smaller ellipses, but they can only draw on half of them at once; you must flip them over to create the entire ellipse.

Conclusion

 

Elliptical Trammel is a double slider crank mechanism that is kinematically inverted and used to draw ellipses with various major and minor axes. Again, there are three such locations that produce various motion traces. The point at the pin joint of both sliders produces a straight motion, whereas a competitive circle is produced if the point is precisely in the center of the crank. The elliptical trammel is regarded as an RRPP mechanism since it has two rotating pairs, two sliding pairs, and a fixed link that serves as the mechanism's framework. The sliding pairs are formed by the slider, while the rotating pairs are created by the connecting rod. As a result, an elliptical trammel is used to mimic a straight motion here.

References

1. https://en.wikipedia.org/wiki/Trammel_of_Archimedes

1. Dr. R.K. Pohane, Sandip N. Pustode, Rahul D. Padole, Swapnil M. Bankar, Rohan Naranje, Swapnil T. Bante “Design and Fabrication of Elliptical Trammel” International Research Journal of Engineering and Technology (IRJET) Vol. 6, issue 3 March 2019