Subject: Kinematics of Machinery

Topic: Kinetics of Rigid Bodies

Difficulty: High

It is an instrument used for drawing ellipses. this inversion is obtained by fixing the slotted plate (link 4) as shown in fig.. The fixed plate or link 4 has two straight grooves cut in it, at right angles to each other. The link 1 and link 3, are known as sliders and form sliding pairs with link 4. The link AB (link 2) is a bar which forms turning pair with link 1 and 3.

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). A little consideration will show that AP and BP are the 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 $\theta$ with the horizontal as shown in fig. (b). Now the co-ordinate of the point P on the link BA will be

$x=PQ=AP\cos\theta\space and\space y=PR=BP\sin\theta$ $\quad\quad\quad\quad\quad\quad\quad$ or

$\frac{x}{AP}=\cos\theta$; $\frac{y}{BP}=\sin\theta$

Squaring and adding

$\frac{x^{2}}{(AP)^{2}}+\frac{y^{2}}{(BP)^{2}}=\cos^{2}\theta+\sin^{2}\theta=1$

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.

• • If the mid-point of link BA, then AP=BP. The above equation can be written as

$\quad\quad\quad\quad\quad\quad\quad$ $\frac{x^{2}}{(AP)^{2}}+\frac{y^{2}}{(BP)^{2}}=1\space or\space{X^{2}}+Y^{2}=(AP)^{2}$

This is the equation of a circle whose radius is AP. Hence if P is the mid-point of link BA, it will trace a circle.