Burmester Curves Research Articles

Synthesis of Four-Bar Linkages by Four Infinitely Close Relative Positions and Pressure Angle

A computer-applicable linear mathematical model has been developed to determine Burmester’s curves for infinitely close relative positions (cubic of stationary curvature), which indirectly uses Carter-Hall’s circle. By varying a free parameter and using elements of kinematic and analytical geometry, an incomparably simpler solution is achieved than that obtained by the third-degree equations of the Burmester's curves for stationary curvature. The mathematical model for the synthesis of four-bar linkages includes and a condition for the pressure angle, whereupon is uniquely defined the kinematic diagram of the mechanism. Of the pressure angle, the reactions of the forces in the kinematic pairs and the force sizing of the mechanism depend. The model would facilitate the engineers in the synthesis of four-bar linkages by generating a function approximating a given function in the vicinity of a given position, where the two functions have four infinitely close common points (3rd-order approximation). An example of the synthesis of a four-bar linkage illustrates the application of the model, which is linear - it includes only equations of straight lines written in Cartesian coordinates, which is why it is convenient for computer calculations.

Analytical synthesis methodology of RCCC linkages for the specified four poses

A new RCCC linkage synthesis theory and methodology for four poses is proposed. A CC dyad usually has double infinite solutions for four specified poses, while an RC dyad usually has no solution. This paper proposes an analytical synthesis methodology by which infinite exact solutions of RCCC linkages can be obtained. Compared with the optimization method proposed by other researchers and iterative method proposed by the authors previously, the analytical synthesis methodology is possible to obtain more accurate solutions in a very short time. The first step is to set up the equations of Burmester curve and all potential solutions that meet the four given orientations can be obtained. The solutions with defects can be eliminated in this step. The second step is to solve remaining equations to obtain a discriminant. Finally, all potential solutions are substituted into the discriminant to determine if the solution is a RCCC linkage solution. The solutions of RCCC linkage are displayed on a map. Two illustrative examples are included.

A unified synthesis method based on solution regions for four finitely separated and mixed “Point-Order” positions

Abstract For rigid-body guidance mechanism synthesis reaching four specified task positions, Burmester curve equations are presented by a displacement matrix method. These equations are suitable for both four-finite and four-mixed position cases. With the transformation of coordinates, and based on the projective geometry theory that a series of intersections of two pencils-of-low-order curves constitute a high-order curve, Burmester curves can be expressed by the intersections of a pencil-of-circles and a pencil-of-lines. Then the expressions of Burmester points are parameterized with the angle φ by employing a solution region method. Therefore, Burmester curves can be single-valuedly and orderly expressed, and an infinite number of mechanism solutions can be expressed on a finite coordinate plane.

A Method of Type Map for Rigid-Body Guidance Based on Burmester Curve

The Burmester curve plays an important role in rigid-body guidance of four-bar linkage, but it still had serious blindness because two center points or circle points should be assigned arbitrary. So a method of Type Map for rigid-body guidance was proposed in this paper. All the four-bar linkages synthesized by burmester curve can be classified in accordance to their attributes. Using this method, satisfactory four-bar linkages can be obtained quickly for the given four rigid-body positions. An algorithm was proposed to define the discrimination of the Type Map in depth. Based on the platform of Matlab, a program was developed to realize the automatic design and visualize the guidance synthesis for four-bar linkage. The result proved the method to be feasible and effective.

A Simple Synthesis Method for Spherical 4R Linkages Reaching Four Positions

For spherical 4R linkage synthesis reaching four specified task positions, we introduce a simple derivation method of spherical Burmester curve equation by employing a displacement matrix method. Then we presented a method to calculate the coordinates of circle and center points, so the spherical Burmester curves can be drawn by the software developed.

On mechanism design optimization for motion generation

A set of Burmester curves can represent an infinite number of planar four-bar motion generator solutions for a given series of prescribed rigid-body poses. Unfortunately, given such a vast number of possible mechanical solutions, it is difficult for designers to arbitrarily select a Burmester curve solution that ensures full link rotatability, produces feasible transmission angles and is as compact as possible. This work presents an algorithm for selecting planar four-bar motion generators with respect to Grashof conditions, transmission angle conditions and having the minimal perimeter value. This algorithm has been codified into MathCAD for enhanced analysis capabilities and ease of use. The example in this work demonstrates the synthesis of a compact planar, four-bar crank rocker motion generator with feasible transmission angles.

Computer modelling and geometric construction for four-point synthesis of 4R spherical linkages

In this paper, computer modelling and geometric construction of Burmester curve for synthesis of spherical mechanisms is presented. Rigid shell guidance in four specified positions on a sphere is performed by a 4R spherical linkage. The synthesis of such a linkage requires obtaining the associated Burmester curves on the reference sphere. Based on the Burmester theory and beside the computational and modelling abilities of the symbolic mathematical software namely Maple, an accurate as well as fast procedure for geometric construction of Burmester curve is developed. In the first part, the concepts of orientation and position on the sphere, pole of the motion and the related parameters are extended for modelling by the Maple. In the second part, the concepts of complementary axis quadrilateral and its imaginary motion, the center and circle axis cones and Burmester curve are derived. In the final part, using the prepared procedure and through a numerical example, a 4R spherical linkage for guiding an antenna to meet four specified postures in a three-dimensional working space is synthesized.

A Method for Adjustable Planar and Spherical Four-Bar Linkage Synthesis

This paper describes a new method to synthesize adjustable four-bar linkages, both in planar and spherical form. This method uses fixed ground pivots and an adjustable length for input and output links. A new application of Burmester curves for adjustable linkages is introduced, and a numerical example is discussed. This paper also compares a conventional synthesis method (nonadjustable linkage) to the new method. Nonadjustable four-bar linkages provide limited solutions for five-position synthesis. Adjustable linkages generate one infinity of solution choices. This paper also shows that the nonadjustable solutions are special cases of adjustable solutions. This new method can be extended to six position synthesis, with adjustable ground pivots locations.

Synthesis of Spherical Four-Bar Mechanisms for Two or Three Prescribed Coupler-Curve Cusps

This paper presents a new approach for the synthesis of spherical four-bar mechanisms with two or three prescribed coupler curve cusps. According to the configuration of a spherical four-bar mechanism whose coupler point coincides with the instant center at two separated positions, two kinds of degenerated curves for the corresponding spherical Burmester curves are obtained. For each degenerated curve, a synthesis procedure is proposed for the dimensional synthesis of spherical four-bar mechanisms to trace a coupler curve with two prescribed cusps. The possible types of spherical four-bar mechanisms with two-cusp coupler curves are discussed. Furthermore, special spherical double-rockers are also presented for tracing a symmetric coupler curve with two cusps, tracing a doubly symmetric coupler curve with two cusps or tracing a symmetric coupler curve with three cusps.

An Introduction to Burmester Field Theory

This paper introduces the Burmester field for motion-generation dyads with four design positions. The Burmester field is the region swept by a Burmester curve as one or more of the design positions varies. The geometric features of the Burmester field are discussed and shown to be related to the poles. The most significant feature are anchor poles that remain stationary as the design positions are changed. The envelope of the Burmester field found with the variation of a single design parameter is developed. This work demonstrates that the envelope of the Burmester field consists of segments of Burmester curves and segments found using envelope theory. A number of examples are presented and discussed. [S1050-0472(00)00501-8]

On Explicit Equations of the Burmester Curves for the PPP-P Case in Spherical Motion

In this paper, an analytic treatment to investigate the Burmester curves of the PPP-P case in spherical motion is presented. With a rotation angle of a coupler about a pole and an instant centre associated with an inflection point specified, a geometric construction is established to determine the corresponding spherical centre-points and circle-points. According to the geometric construction, the explicit equations of spherical Burmester curves for the PPP-P case are directly derived by using spherical trigonometry, without the use of differential technique. The explicit equations obtained give a clear insight into the geometric properties and special cases of the spherical Burmester curves, which include degenerated curves and symmetric curves. A design application is presented.

EXPLICIT EQUATIONS AND SPECIAL CASES OF THE BURMESTER CURVES FOR THE PPP-P CASE

This paper presents an analytic treatment to investigate the Burmester curves of the PPP-P case. Provided that an instant center associated with an inflection point and rotation angle of a coupler about a pole are specified, the explicit equations of Burmester curves for the PPP-P case are directly derived without any differential process. Because the explicit equations derived are of quadratic form, their geometric properties can be easily investigated. Six degenerated cases and three special cases of the Burmester curves for the PPP-P case are obtained. A design example is presented.

THE DEGENERATED SPHERICAL BURMESTER CURVES OF THE PP-P-P CASE

In this paper an analytic method to investigate the degenerated spherical Burmester curves of the PP-P-P case is presented. (Provided that an instant centre and two poles associated with two rotation angles of a coupler are specified, a geometric construction is established to determine the spherical centre- and circle-point.) According to the geometric construction, the explicit equations of spherical Burmester curves for the PP-P-P case are thus derived, without the use of differentiation, but by using spherical cross ratio. By assigning proper parameter values to equations obtained, six types of degenerated spherical Burmester curves of the PP-P-P case are obtained directly, which include three different forms of the degenerated curve: (1) a point, (2) a great circle and a spherical Thales ellipse and (3) three great circles.

Synthesis of spherical four-bar path generator satisfying the prescribed tangents at two cusps

Design equations for spherical four-bar linkages to trace a coupler curve with two prescribed cusps are derived in this study by using a special case of spherical Burmester curves. For the case in which the tangents at two cusps are also prescribed, an analytical method is proposed with the aid of the foregoing design equations associated with the concept of the spherical cross ratio and spherical Bobillier theorem. The proposed method is straightforward and quite useful for those cases in which the tangents at two cusps are neither on the same plane nor on parallel planes. Numerical examples are also provided for illustrating the entire technique.

DEGENERATE BURMESTER CURVES OF PP-PP MULTIPLY SEPARATED POSITIONS

A new method is proposed to derive the special cases that the center- and circle-point curves of PP-PP multiply separated positions degenerate into the combinations of points, lines and circles. Analytical equations for the construction of the Burmester curves are presented in a simpler form with four parameters. All conditions that yield degenerate forms of the equations are concluded by investigating the parameters. Degenerate Burmester curves are then thoroughly derived from various combinations of the conditions. The results of this work are more complete than existing papers, and presented concisely in tables. Applications of the degenerate curves are also illustrated by an example.

Circuit Rectification for Four Precision Position Synthesis of Stephenson Six-Bar Linkages

Circuit rectification ensures that a linkage may reach all precision positions without disassembly. Circuit rectification is crucial to the practical synthesis of Stephenson linkages, which have up to six circuits. Causes of circuit defects that may arise during synthesis are quantified first. A procedure is then developed to identify the segments of Burmester curves which produce mechanisms with all precision positions on the same circuit. The procedure is applicable to any Stephenson linkage.

Six and Seven Position Triad Synthesis Using Continuation Methods

A continuation method is used for the synthesis of triads for motion generation with prescribed timing applications. The procedure is applied to solve both six and seven position synthesis problems. Triad Burmester curves are generated for the six position synthesis problem and an eight-bar mechanism is designed to illustrate the procedure. For the seven position synthesis problem, a finite number of solutions are obtained. A geared five-bar, seven position path generation example is considered.

Order Rectification for Complex Number Based Burmester Curves

A new solution to the order rectification problem for a driving dyad of planar mechanisms is presented. The method identifies sections of both Burmester curves where the driving link rotates in a single direction when passing through four precision positions in sequence. The new solution describes desirable regions of the curves in terms of the complex number parameters used to generate the curves, providing a complex number equivalent to available pole based order rectification procedures. The new solution is stated in a summary form that is readily codifiable. An example is presented. The theory underlying the new solution is then developed in detail.

Three-Position Function Generation of Planar Four-Bar Mechanisms With Equal Deviation Transmission Angle Control

This paper is an extension of earlier work on mapping of three-position function generation of planar four-bar mechanisms. Previously, it has been shown that all of the potential solutions to a given problem may be represented in an αβ-plane which can be subdivided into mechanism types. Further, the regions in the αβ-plane may represent two possible forms of assembly plus a change of form class which are not valid solutions. In this paper, we provide a third-order polynomial which defines the locus in the αβ-plane of solutions which have equal deviation of their transmission angle from the ideal of 90° throughout the entire range of motion. When these solutions are mapped into a Cartesian plane, the ground pivot locations produce curves similar to the familiar Burmester curves for four-position synthesis problems. Additional advantages of the approach are that the input link is automatically a crank, the desired link length ratio can be controlled, and the solutions are free of defects.

An algebraic treatment of Burmester points by means of three basic poles

In guiding a body through four finitely separated coplanar positions, an algebraic method for computing the Burmester curves, i.e. the centre-point and circle-point curves, is developed on the basis of three basic poles. This method is then extended to other four-position cases, including finitely and infinitesimally separated positions. A five-position case is considered as a combination of two four-position cases. The final equation is then a quartic equation which contains no extraneous roots other than the four required Burmester-centres (Burmester points). Examples are given.