The Garden of Archimedes
 The Garden of Archimedes
 A Museum for Mathematics

Curves and mechanisms

F. Conti

(from the catalogue of the exhibition)

What is a mechanism ?

The problem of rectilinear motion without friction

Watt's mechanism

Tchebycheff's mechanism

Peaucellier's inverter

Hart's connecting rod mechanism

Curves, connecting rod mechanisms and profiles

The articulated quadrilateral and some applications

Curves, connecting rod mechanisms and profiles   

Algebraic curves and connecting rod mechanisms

In 1875 Kempe demonstrated that any algebraic curve can be drawn with a connecting rod mechanism. But also for a simple curve like a conic section, the mechanism can be very complicated.
In the movement of a horizontally pivoted door, like the ones of many garages, the point X describes an ellipse, but it would be out of place to state that the mechanism was realised for this purpose.

Profiles and curves

On the contrary, in technology it sometimes occurs that it is the profile of a certain curve that determines the functions of a mechanism.
Just one example is enough: to evenly wind a thread around a spool it is necessary that the slide that guides the thread moves with constant speed (alternatively from right to left and from left to right). If the simplest possible mechanism was employed, with a slide of the thread moving circularily and evenly, the projection of such slide on the axle of the spool (which is the position of the winding thread) would move harmoniously, placing a lot more thread at the extremities than at the centre.
A solution to the problem can be obtained using the spyral of Archimedes. This is a curve where the points P have the following property: their distance from a fixed point O increases proportionally to the angle formed by the segment PO with a reference fixed half-line OA. It is a curve that is often encountered in everyday use objects (magnetic tapes, roll of ropes, the decorations of certain cakes ...).
Using a cam formed by two "block" of the spiral of Archimedes it is very easy to solve the problem of winding the thread evenly. This is the solution universally adopted to wind the "spool " in sewing machines.


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