Linkages

What are linkeages?

image031.jpg
The three bar linkage in Figure 1 reverses motion.
http://kmoddl.library.cornell.edu/linkages/

In general, if you have a collection of straight sticks hinged to one another at the ends, then you can say you have a linkage.






The slider-crank linkage, which converts linear to rotary motion and vice versa, is one of the most widely used mechanisms in the world due to its use in most of the internal combustion engines in the form of a crankshaft, piston-cylinder and connecting rod.




Linkages can be found in many places, like the windshield wipers and in lamps:
  
Robot arms are another example of linkages – in fact, it is possible that people started to think about the use of linkages because of their similarity to our own arms.



The pantograph is based on geometrical proportions that were known since ancient times. It can be called the earliest copying machine, making exact duplicates of written documents. Artists adopted its use for duplicating drawings

Ordinary pantograph. http://www.daube.ch

On a recent visit to Charlottesville, VA, I took a tour of Monticello and saw Jefferson's pantograph. He acquired his letter-copying device in 1804. It was based on the principles of the pantograph, and had two pens that were moved simultaneously by the writer's hand, making a duplicate copy of the original.



  • Point O is fixed
  • All bars have a constant length
  • The structure pivots at L, M, N
  • A pointer at Q traces over the shape being drawn
  • The pen at P reproduces the shape. In this case on a larger scale.
  • Figure LMQN is a parallelogram.
  • Points O, P, Q are collinear—in a line.


If you were making a model, you might find it easier to place pointer Q away from the pivot. In this case you would need to adjust the position of P to retain the collinearity of points O, P, Q.

We easily see that the pantograph produces a mathematical enlargement. For by similar triangles ONQ, OLP, OQ : OP = ON : OL = a : a + c which is constant. If c = a the pantograph would double the size of the original figure.



Also check out MoMath's Math Monday: Linkages – Biomimicry by Glen Whitney