ELLIPTICAL TRAMMEL

IntroductionThe 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.

IntroductionAs 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.

InversionElliptical 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

InversionPair 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.

MechanismThis 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.

MechanismWhen 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 :

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

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.

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.