Burmester Curve

[Willy Aucoin]

Burmesterbegan 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.[2] 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.[3][4][5] The approach has been generalized to the synthesis of spherical and spatial mechanisms.[6]

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 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.

One of the more common applications of a four-bar linkage appears as a rod that connects two levers, so that rotation of the first lever drives the rotation of the second lever. The levers are hinged to a ground frame and are called the input and output cranks, and the connecting rod is the called the coupler link. Burmester's approach to the design of a four-bar linkage can be used to locate the coupler so that five specified angles of the input crank result in five specified angles of the output crank.

This formulation of the input-output synthesis of a four-bar linkage is an inversion of finite-position synthesis, where the movement of the output crank relative to the input crank is specified by the designer. From this viewpoint the ground link OC is a crank that satisfies the specified finite positions of the movement of the output crank relative to the input crank, and Burmester's results show that its existence guarantees the presence of at least one coupler link. Furthermore, Burmester's results show that there may be as many as three of these coupler links that provide the desired input-output relationship.[6]

The most popular set of such rulers is made up of 3 rulers chosen among a set of 28 that were described in a book by Burmester at the beginning of the XXth century. (Lexicon der Gesamten Technik (1904)). For this reason they are often known also as Burmester's curve.

There seems to be little secondary literature on this so answering the OP questions fully would take some serious digging into the original sources. One promising secondary source that I was unable to locate is Ceccarelli and Koetsier, Burmester and Allevi: A Theory and its Application for Mechanism Design at the end of 19th Century, Proceedings of IDETC/CIE 2006 ASME 2006 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference. I will only give some comments.

Burmester's doctoral thesis was on the geometry of isophotes (lines of equal brightness), but his opus magnum Lehrbuch der Kinematik (Textbook of Kinematics, 1888) developed kinematic synthesis of curve-drawing linkages, an elegant and sadly forgotten theory originated by Reuleaux (Gibson has a brief section on it in his Elementary Geometry of Algebraic Curves (1998), which ironically does not even mention Reuleaux). Roughly, the idea was to sketch various curves by designing linkages and other mechanical devices whose points would trace them in motion, such as the classical four-bar linkage employed in Watt's engine. It seems likely that the curves Burmester used were linkage coupler curves (traced by a point on the coupler bar). The cubic of stationary curvature (an asymmetric strophoid, below), traced by the four-bar linkage, is reminiscent of the shapes in French curves. Hartenberg's Kinematic Synthesis of Linkages, published when the subject was still lively (1964), gives some history:

"Modern kinematics had its beginning with Reuleaux. His now classical "Theoretische Kinematik" of 1875 presented many views finding general acceptance then that are current still... Reuleaux regarded a mechanism as a (kinematic) chain of connected links (or parts), one link being fixed... Through hindsight we saw in Watt's straight-line linkage the amorphous beginning of an ordered and advanced synthetic process. Nearly a century later Reuleaux identified synthesis as a concept, as an entity that might be and must be pursued to guide the designer through the maze of mechanisms. His views were limited to only type synthesis: by this is meant the determination of the type of mechanism for a given job... which lies ahead of the related and more recent fields of number and dimensional synthesis.

[...] Geometers and algebraicists of the 1870s became interested in linkages as curve-drawing devices, not as hardware; their work has since been made part of the corpus of the kinematics of mechanisms. It was discovered that a link motion can be found to describe an algebraic curve of any order. In particular, the coupler-point motion of the four-bar linkage (then called a "three-bar motion") was studied. Samuel Roberts showed that a coupler point describes a curve of the sixth order.

[...] Concepts far beyond those of Reuleaux were added to the picture he had attempted to paint; the whole is now called synthesis of linkages... The third and last kind is dimensional synthesis, or the determination of the proportions (lengths) of the links needed to accomplish the specified motion transformation. In Germany, Burmester was in accord with Reuleaux's fundamental concepts and most of his nomenclature and definitions. Making extensive use of mathematical principles (mostly geometrical), and considering displacement, velocity, and acceleration, Burmester's "Lehrbuch der Kinematik" (1888) developedgeometric methods that furthered analysis and showed the way to synthesis.

A French curve is a template usually made from metal, wood or plastic composed of many different curved segments. It is used in manual drafting and in fashion design to draw smooth curves of varying radii. The curve is placed on the drawing material, and a pencil, knife or other implement is traced around its curves to produce the desired result. They were invented by the German mathematician Ludwig Burmester and are also known as Burmester (curve) set.

French curves are used in fashion design and sewing alongside hip curves, straight edges and right-angle rulers. Commercial clothing patterns can be personalized for fit by using French curves to draw neckline, sleeve, bust and waist variations.

The arrangement of ligament fibers in a parallel manner such as the medial collateral ligament (MCL), fibular collateral ligament (FCL), or a tendon provides a structure with a high maximum load to failure because the majority of the fibers are loaded in a symmetrical manner and share the loading profile. The structure will fail at a relatively small displacement, as is true for the MCL and FCL, which are discussed in Chapter 2, Lateral, Posterior, and Cruciate Knee Anatomy. Alternatively, the capsular structures are designed to provide a low stiffness, allowing elongation or compliance, which is built into the microgeometry similar to that shown in the illustration of the finger-trap in Figure 4-5. Certain portions of the capsule may have an arrangement in which some fibers exist in a more parallel array, such as the posterior oblique ligament of the posteromedial capsule. Failure of capsular structures can be difficult to detect because a microtearing process occurs throughout the entire capsule. The capsule may appear slack and elongated without an obvious failure or the capsule may be avulsed off of its femoral or tibial attachment and the failure site identified and repaired at surgery. Prior to ultimate failure of the capsule, the microfailure process produces a residual elongation or slackening that, in chronic knee injuries, may require plication procedures to restore tension and function. An example is plication of posteromedial or posterolateral capsular structures so that they will function in full extension and resist knee hyperextension. The length-tension behavior and failure pattern of the ACL and PCL are also dissimilar, with different fiber regions brought into the loading based on the position of the knee.

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