Elliptical trammel

1. ELLIPTICAL TRAMMEL

2. 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 .

3. 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.

4. 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

5. 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.

6. 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.

7. 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 :

8. 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

9. 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.

10. 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.