Understanding Elliptical Trammels and Mechanism Inversions

An elliptical trammel is a fascinating mechanical device that draws, cuts, or mills ellipses. Its operation is based on the principle of mechanism inversions, specifically derived from a fundamental kinematic chain.

What is a Double Slider Crank Chain?

A double slider crank chain is a variation of the basic slider crank chain. It consists of four kinematic links. In this mechanism:

• Two links are sliders.
• Two links are turning pairs (revolute joints).
• There are two sliding pairs.

The links include a frame (which is stationary), two sliders that move along perpendicular paths, and a connecting rod joining the two sliders.

Mechanism Inversions Explained

An inversion of a kinematic chain is obtained by fixing a different link in the chain. Each inversion results in a mechanism with different relative motions between the links, leading to various useful applications.

Elliptical Trammel as an Inversion

The elliptical trammel is created by fixing one of the links of the double slider crank chain. When the link connecting the two sliders (the equivalent of the connecting rod) is taken as the coupler link, and the two sliders are constrained to move along perpendicular slots (the equivalent of the fixed links), the path traced by any point on the connecting rod is an ellipse. If the fixed link is considered the frame, and the sliders move along the frame, the motion of a point on the connecting rod traces the ellipse.

Let's consider the links of a double slider crank chain:

1. Link 1: Fixed link (frame, e.g., two perpendicular guides/slots).
2. Link 2: Slider moving in one guide.
3. Link 3: Connecting rod joining the two sliders.
4. Link 4: Slider moving in the other guide.

In the case of the elliptical trammel, the frame (Link 1) is typically the two perpendicular guides. Link 2 slides in one guide, Link 4 slides in the other guide, and Link 3 is the rod connecting them. A point on Link 3 traces an ellipse. This is a direct application resulting from fixing the frame link in the double slider crank chain.

Why Not Other Options?

• Quick Return Mechanism: Mechanisms like the Whitworth or oscillating cylinder quick return mechanisms are typically inversions of a single slider crank chain, designed to provide a slower cutting stroke and a faster return stroke. They involve one sliding pair and three turning pairs.
• Single Slider Crank Chain: This basic mechanism has one sliding pair and three turning pairs. Its inversions include the reciprocating engine mechanism, crank and slotted lever mechanism, and oscillating cylinder engine mechanism. It does not directly form the basis for the elliptical trammel.
• Four Bar Chain: This is a mechanism with four links connected by four turning pairs. Its inversions include beam engine, coupling rod (locomotive mechanism), and Watt's indicator mechanism. It is fundamentally different from mechanisms involving sliders.

Therefore, the elliptical trammel is specifically an example of an inversion of the double slider crank chain.