Elliptical trammel


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Elliptical trammel

  1. 1. Introduction The Elliptical Trammel (also known as the Elliptic Trammel, or the Trammel of Archimedes) is a simple mechanism which can trace an exact elliptical path. It consists of two shuttles which are confined (trammelled) to perpendicular channels or rails, and a rod which is attached to the shuttles by pivots at fixed positions along the rod.
  2. 2. Introduction As the shuttles move back and forth, each along its channel, the end of the rod moves in an elliptical path. The semi-axes a and b of the ellipse are the distances between the end of the rod and the two pivots. An ellipsograph is a trammel of Archimedes intended to draw, cut, or machine ellipses, e.g. in wood or other sheet materials. Usually the distances a and b are adjustable, so that the size and shape of the ellipse can be varied.
  3. 3. Inversion Elliptical Trammel is an inversion of double crank chain mechanism. A double slider crank chain consists of four links forming two sliding pairs and two turning pairs. Link 1 Frame Link 2 Slider –I Link 3 Coupler Link 4 Slider - II
  4. 4. Inversion Pair A- Sliding pair, Link 1 and Link 4. Pair B- Turning pair, Link 1 and Link 2. Pair C-Turning pair, Link 2 and Link 3. Pair D-Sliding pair, Link 3 and Link 4.
  5. 5. Mechanism This inversion is obtained by fixing link 4. The mechanism so obtained is called elliptical trammel which is shown in Figure . This mechanism is used to draw ellipse. The link 1, which is slider, moves in a horizontal slot of fixed link 4. The link 3 is also a slider moves in vertical slot.
  6. 6. Mechanism When the links 1 and 3 slide along their respective grooves, any point on the link 2 such as P traces out an ellipse on the surface of link 4, as shown in Fig. A little consideration will show that AP and BP are the semi-major axis and semi-minor axis of the ellipse respectively. This can be proved as follows :
  7. 7.  Let us take OX and OY as horizontal and vertical axes and let the link BA is inclined at an angle with the horizontal, as shown in Fig. Now the co-ordinates of the point P on the link BA will be x = PQ = AP cos t and y = PR = BP sin t x/AP = cos t Similarly, y/BP = sin t
  8. 8.  On Squaring and Adding , we get, This is the equation of an ellipse. Hence the path traced by point P is an ellipse whose semi- major axis is AP and semi-minor axis is BP.
  9. 9. Applications It is used in automatic tool changer in a machining. Elliptical Trammels are used for drawing large ellipses. They can be used to draw smaller ellipses but only draw one half at a time, having to be reversed to draw the complete ellipse.