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Moving Linked Rods

Many mechanical components rely on linkages of rods that form a mechanism that can move in certain ways, partly constrained and partly flexible. Such mechanisms are often used to transfer power to drive machine parts. The bars that drive the wheels on a steam locomotive are a common example, as are the arms that control dentists' drills. In flat linkages, the bars are joined by bolts or slip joints that constrain the motion to two dimensions. In other linkages, some joints rotate around an axis, thus permitting motion in three dimensions.

A flat linkage of four bars in the form of a quadrilateral will flex in two dimensions, but once the lengths of the bars are fixed, only certain motions are possible. One use of such a linkage is to transform circular to oscillating motion. Suppose A is the center of a motor that turns rod AB of a quadrilateral linkage ABCD in a circular motion. Vertex D is also fixed, so as B moves in a circle around point A, C moves along a path determined by the lengths of the four rods. In one particular application, the fixed base AD is 5 feet and the rotating arm AB is 1 foot long. What motions are possible for point C for various lengths of rods BC and CD?

Suppose a fifth rod is added, making a flexible pentagon with vertices ABCDE. A and E are located at the centers of motors that turn rods AB and ED in circular motions. If AE is 5 feet, and AB and ED are 1 foot each, what happens to point C as AB and ED turn at different rates?

Comment: Workplace examples such as linkage problems are quite common. They involve simple mechanisms that turn, slide, and rotate to make things go. Anyone who has tried to clear a paper jam from the inside of a Xerox machine has seen multiple examples of mechanisms at work. Anyone who tries to fix such a machine needs to understand how geometry serves the cause of making things move. Problems of this type exercise geometric thinking in powerful ways. They cry out for physical models, which are easy to build. Many variations are possible, and all have good physical applications. The mathematics involves applications of coordinate geometry as well as trigonometry and simple algebra. Much of the analysis can be carried out either by geometry, analytic geometry, algebra, or spreadsheets.

Excerpted from "Mathematics for Work and Life" by Susan L. Forman and Lynn Arthur Steen (in Seventy-Five Years of Progress: Prospects for School Mathematics, NCTM, 1995).

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Last Update: 12/29/98