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- W261340738 abstract "Four bar linkages (sometimes called three bar linkages) have been investigated by mathematicians, engineers and artists for over three hundred years. A natural extension to three dimensions generates a family of aesthetically interesting surfaces. The Lemniscate The plane curve known as the lemniscate was first described by Jakob Bernouilli in 1694 (in Acta Eruditorium). There are several elegant ways to construct this curve [1], perhaps the best known is by means of a four-bar linkage, known as Watt’s linkage. This was widely used in steam-driven beam engines to generate approximate straight line motion [2]. Two bars of unit length pivot around points that are fixed √2 apart (by a notional third bar). They are joined by a fourth bar, length √2, and the mid-point of this bar moves on a lemniscate, while its ends describe unit circles (figure 1). John Sharp has written an article that describes some less well-known properties of this curve [3]. Figure 1: The lemniscate of Bernouilli as a locus of a point on a four bar linkage. More General Curves It is easy to see that this arrangement can actually generate a five parameter (one is a scale factor) family of curves by altering the lengths of the four bars and the position of the drawing point on the fourth bar (possibly extended). Computer geometry packages provide a convenient way to investigate these curves, and there are several interactive animations available on the internet, for example [4]. The drawing point need not lie on the fourth bar but could be fixed to it some distance away, an idea that was used by Daniel Wood, an architectural mason, at the end of the nineteenth century [5]. Figure 2 shows one of the plates from his book. The pivoting bars are strings, each labelled S, the fourth bar is a triangle, labelled T, and the drawing point is labelled P.The idea was developed further by Richard S. Hartenberg, a professor of mechanical engineering, who co-authored a book on linkages [6], and published a paper in Leonardo describing" @default.
- W261340738 created "2016-06-24" @default.
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- W261340738 date "2008-01-01" @default.
- W261340738 modified "2023-09-24" @default.
- W261340738 title "Four Bar Linkages" @default.
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