Burmester theory

Burmester theory is named after Ludwig Burmester (1840–1927). Burmester introduced geometric techniques for synthesis of linkages in the late 19th century. His approach was to compute the geometric constraints of the linkage directly from the inventor's desired movement for a floating link. From this point of view a four-bar linkage is a floating link that has two points constrained to lie on two circles.

Burmester began with a set of locations, often called poses, for the floating link, which are viewed as snapshots of the constrained movement of this floating link in the device that is to be designed. The design of a crank for the linkage now becomes finding a point in the moving floating link that when viewed in each of these specified positions has a trajectory that lies on a circle. The dimension of the crank is the distance from the point in the floating link, called the circling point, to the center of the circle it travels on, called the center point. Two cranks designed in this way form the desired four-bar linkage.

This formulation of the mathematical synthesis of a four-bar linkage and the solution to the resulting equations is known as Burmester Theory. The approach has been generalized to the synthesis of spherical and spatial mechanisms.

Burmester theory seeks points in a moving body that have trajectories that lie on a circle called circling points. The designer approximates the desired movement with a finite number of task positions; and Burmester showed that circling points exist for as many as five task positions. Finding these circling points requires solving five quadratic equations in five unknowns, which he did using techniques in descriptive geometry. Burmester's graphical constructions still appear in machine theory textbooks to this day.

Two positions: As an example consider a task defined by two positions of the coupler link, as shown in the figure. Choose two points A and B in the body, so its two positions define the segments A¹B¹ and A²B². It is easy to see that A is a circling point with a center that is on the perpendicular bisector of the segment A¹A². Similarly, B is a circling point with a center that is any point on the perpendicular bisector of B¹B². A four-bar linkage can be constructed from any point on the two perpendicular bisectors as the fixed pivots and A and B as the moving pivots. The point P is clearly special, because it is a hinge that allows pure rotational movement of A¹B¹ to A²B². It is called the relative displacement pole.

Three positions: If the designer specifies three task positions, then points A and B in the moving body are circling points each with a unique center point. The center point for A is the center of the circle that passes through A¹, A² and A³ in the three positions. Similarly, the center point for B is the center of the circle that passes through B¹, B² and B³. Thus for three task positions, a four-bar linkage is obtained for every pair of points A and B chosen as moving pivots.

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