Burmester's theory

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.

P is the pole of the displacement of A¹B¹ to A²B²

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 or also instant centre of rotation.

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.

Four positions: Graphical solution to the synthesis problem becomes more interesting in the case of four task positions, because not every point in the body is a circling point. Four task positions yield six relative displacement poles, and Burmester selected four to form the opposite pole quadrilateral, which he then used to graphically generate the circling point curve (Kreispunktcurven). Burmester also showed that the circling point curve was a circular cubic curve in the moving body.

Five positions: To reach five task positions, Burmester intersects the circling point curve generated by the opposite pole quadrilateral for a set of four of the five task positions, with the circling point curve generated by the opposite pole quadrilateral for different set of four task positions. Five poses imply ten relative displacement poles, which yields four different opposite pole quadrilaterals each having its own circling point curve. Burmester shows that these curves will intersect in as many as four points, called the Burmester points, each of which will trace five points on a circle around a center point. Because two circling points define a four-bar linkage, these four points can yield as many as six four-bar linkages that guide the coupler link through the five specified task positions.

Burmester's approach to the synthesis of a four-bar linkage can be formulated mathematically by introducing coordinate transformations [T_i] = [A_i, d_i], i = 1, ..., 5, where [A] is a 2×2 rotation matrix and d is a 2×1 translation vector, that define task positions of a moving frame M specified by the designer.[6]

The goal of the synthesis procedure is to compute the coordinates w = (w_x, w_y) of a moving pivot attached to the moving frame M and the coordinates of a fixed pivot G = (u, v) in the fixed frame F that have the property that w travels on a circle of radius R about G. The trajectory of w is defined by the five task positions, such that

${\displaystyle \mathbf {W} ^{i}=[T_{i}]\mathbf {w} =[A_{i}]\mathbf {w} +\mathbf {d} _{i},\quad i=1,\ldots ,5.}$

Thus, the coordinates w and G must satisfy the five equations,

${\displaystyle (\mathbf {W} ^{i}-\mathbf {G} )\cdot (\mathbf {W} ^{i}-\mathbf {G} )=R^{2},\quad i=1,\ldots ,5.}$

Eliminate the unknown radius R by subtracting the first equation from the rest to obtain the four quadratic equations in four unknowns,

${\displaystyle (\mathbf {W} ^{i}-\mathbf {G} )\cdot (\mathbf {W} ^{i}-\mathbf {G} )-(\mathbf {W} ^{1}-\mathbf {G} )\cdot (\mathbf {W} ^{1}-\mathbf {G} )=0,\quad i=2,\ldots ,5.}$

These synthesis equations can be solved numerically to obtain the coordinates w = (w_x, w_y) and G = (u, v) that locate the fixed and moving pivots of a crank that can be used as part of a four-bar linkage. Burmester proved that there are at most four of these cranks, which can be combined to yield at most six four-bar linkages that guide the coupler through the five specified task positions.

It is useful to notice that the synthesis equations can be manipulated into the form,

${\displaystyle (\mathbf {W} ^{i}-\mathbf {W} ^{1})\cdot \left({\frac {\mathbf {W} ^{i}+\mathbf {W} ^{1}}{2}}-\mathbf {G} \right)=0,\quad i=2,\ldots ,5,}$

which is the algebraic equivalent of the condition that the fixed pivot G lies on the perpendicular bisectors of each of the four segments Wⁱ − W¹, i = 2, ..., 5.