Coupler Point Research Articles

Vereinfachte bestimmungsgleichungen für bewegungs-weiterleitungen von koppelkurven mit spitzen

Each four-bar linkage running as a crank and rocker mechanism has one unique coupler point producing a coupler curve with two cusps using this special coupler point, complex mechanisms can be designed having exact dwells. In this paper a four-bar linkage is used having the most favorable transmission behavior. Using simple equations, the mechanism and the coupler point location can be calculated. Using a rotating inverted slider output link, the designer can produce two rather long dwells through a full motion period.

Complex Number Synthesis of Four-Bar Path Generating Mechanisms Adjustable for Multiple Tangential Circular Arcs

By combining complex number methods of synthesis [1] with graphical techniques, an adjustable, planar, four-bar path generating mechanism can be synthesized to produce symmetrical coupler curves that conform to a specified radius over a finite range. In this way the earlier works by Tao and Yan [2, 3] can be generalized for use by a large number of mechanism designers. The coupler point of the mechanism will trace any desired number of circular arcs that are tangential at a preselected point with prescribed radii of curvature (positive, negative, or infinite). In this development, components of this mechanism may be chosen freely or under certain parameters by one of three techniques presented here. Also available is an analysis to give coupler link rotation, output crank rotation, and coupler point location as a function of input crank rotation.

Proposals for a finite 5-dimensional atlas of crank-rocker coupler curves

The coupler curves traced by planar 4-bar crank-rocker linkages are examined with the objective of providing guidelines for a new awareness survey. It is found that the coupler points chosen by Hrones and Nelson[1] in the best known atlas of this kind leads to some avoidable duplication and omissions. It is concluded that coupler points should be confined to a circle on the coupler plane having centre on the coupler-rocker bearing and radius equal to the distance between coupler bearings. Five non-dimensional variables are proposed, three of which are discussed in more detail in earlier work[6].

Highest-order coupler points of Watt-1 linkages, tracing symmetrical 6-bar curves

Certain non-symmetrical Watt-1 linkage mechanisms may produce symmetrical curves. The dimensions of such mechanisms then have to meet five precise conditions. They have been derived from the two possibilities that symmetrize the Assembly Configuration containing the four 6-bar curve cognates. Each possibility leads to a different mechanism. The one that can be driven without additional conditions, contains a kite 4-bar, carrying rigid triangles that are similar and otherwise related. Then, a ‘kite-cell’ is involved, transforming the input-circle into a symmetrical Watt-1 curve of the 8th or of the 12th degree, depending on the choice of coupler-plane. The second possibility leads to a mechanism in which not only a point, but also a straight-line reaches the image positions. This type, however, must move through a ‘stretched position’ with a two-fold coupler motion. Then, gear-pairs or the like are necessary to overcome such a bifurcated position.

Verbesserung der genauigkeit und laufgüte sechsglierdriger koppel-schwingrastgetriebe

In order to produce dwell-rise-dwell motions cam mechanisms can be designed much simpler than linkages. Because of the many advantages of linkages, more attention should be spent to their design and at first to the six-linked structures which fulfill the given problem by smallest expense. The six-bar coupler linkage is discussed here. It realizes the dwells by approximation of the coupler-curve to a circle which is represented by the length of an additional coupler link. Many investigations have been made to meet this problem best. So the circle point curve (Burmester-curve) represents a four-fold coincidence of coupler-curve and circle. If a dwell at the end of an oscillating motion is ordered, the curvature of coupler-curve must show an extremum value. If, in addition, two end-dwells are asked for the classic Burmester Theory never can solve this problem. Now the computer-aided design is able to give new ideas as follows: The major size of coupler-curves is well-known by tables like Hrones and Nelson's atlas and also the ranges usable for dwell-motions. Such four-bars are computer-programmed to find maximum values of curvature automatically. Deviations of the transmitting link from radius of maximum curvature enlarge the duration of dwell but increase the errors during the dwell period. In each case a four-fold coincidence is realized. The computer automatically finds all extremum curvature point and there are no difficulties in finding a second end-dwell. Using special four-bars with equal lengths of coupler, output link and coupler point distance, symmetrical coupler curves are produced in which extremum curvatures correspond to crank displacements of 180°. If non-symmetrical motions are wanted, four-bars with general dimensions have to be computer-investigated. It should be stated that by the method explained eight precision points of coupler curve have been found when the maximum of them is nine. For each problem a certain number of usable four-bars can be found. The computer cam then be ordered to find that one having smallest maximum acceleration. Further investigations in this direction should be started in order to find the limits of six-linked mechanisms and to learn exactly for which requirements linkages of higher order have to be used, i.e. how to produce longer dwells and those ones with higher accuracy, also how to produce two dwells having a larger difference in back and forth motion between dwells and finally how to design linkages for more than two dwells.

The synthesis of 4-bar linkage coupler curves using derivatives of the radius of curvature—I. Straight path procedure

A procedure allowing two degrees of freedom in locating a coupler point is presented for the case of an approximately straight coupler point path such that the zeroes of the first and second derivatives of the radius of curvature are approximated. In the case of the two degrees, both the first and second derivatives of the radius of curvature are specified with respect to the loci of the zeroes of the derivatives. For the synthesis procedure, an example is presented illustrating the entire technique. This is Part I, Straight Path Procedure; Part II will present Circular Path Procedure.

The synthesis of 4-bar linkage coupler curves using derivatives of the radius of curvature—II. Circular path procedure

The procedure presented in Part I is extended to the case where a 4-bar linkage may by synthesized such that a point on the coupler will trace an approximately circular portion of its coupler curve. The radius and center of this circular coupler curve is prescribed by the definition of a 4-bar linkage in which one of the revolutes will reflect this characteristic. Through the approximation of the behavior of the zeroes of the derivatives of the radius of curvature of the coupler point paths, a family of 4-bar linkages is identified; each member of the family will have approximately the same coupler motion.

The Displacement Synthesis of Four-Bar Straight-Line Mechanisms

This paper presents an analytical method for synthesizing a four-bar linkage from a slider-crank mechanism using the cubic of stationary curvature of the slider crank. A coupler point on the resulting linkage will generate an approximate straight line, and the functional relationship between the displacement of the coupler point and the crank angle will be the same as the relation of slider displacement to crank angle of the original slider crank.

Approximate Synthesis of Plane Four-Bar Mechanism for Curve Generation

In this report, an analytical method is presented for the synthesis of a plane four-bar mechanism to generate approximately a desired curve. This method produces such a curve generator that the coupler point may pass through precision points on the desired curve in the stationary coordinate system and the higher order differential coefficients of the coupler point coordinates with respect to an input variable may coincide with the ones of the desired curve at each precision point. Moreover, a numerical procedure is worked out for evaluating the difference between a desired curve and a generated curve.

Four-bar synthesis for the coordination of coupler point displacement with the rotations of both cranks

The correlation of the motion of the coupler point of a four-bar linkage with that of both cranks (partially or fully) is of some practical importance as shown by Hain[1] who gave solutions based either on point position reduction or graphical interpolation. Point position reduction, which loses the infinity of possible designs, and graphical interpolation are replaced in the present paper by improved procedures.

Kinematic Synthesis of Spatial Path-Generating Mechanisms

A design technique for coupler curve generating mechanisms is developed. The procedure is based on a multi-variable search using an objective function developed from topological properties of spatial curves in an arbitrary reference system. The method completely specifies the mechanism and its orientation relative to the chosen co-ordinate system, which generates the best approximation to the desired curve. The procedure always converges from any set of starting conditions. The technique is well suited to digital computer design of planar and spatial mechanisms, as it permits both the synthesis and the kinematic analysis of the mechanism. The 4R and the RGCR four-bar spatial linkages with coupler points are used to illustrate the synthesis procedure.

Kurvengetriebe mit umlaufender Doppelkurve zur erzeugung von Bahnkurven mit vorgeschriebener geschwindigkeit

Methods of producing plane curves by linkages have often been described; many papers deal with these. It is surprising that the use of cam-linkages for this purpose is rarely treated. If it is required that a coupler point follow a given closed curve and, in addition, that its velocity at each position be prescribed, two closed cam profiles are necessary. Several papers in the past have shown how to produce a desired curve, but in most the velocities have been neglected. A systematic approach is here used to indicate the structures of all suitable cam-linkages having two closed cams. In this paper only those designs are investigated which have a fully rotating input cam body consisting of two profiles. In order to find the least expensive mechanisms, only four link cam-linkages are considered. In the opinion of the author, here is one of the most important and most serious problems of designers, i.e. the need to solve each problem in the best way and also with a minimum of expense. There are many difficulties in doing this. In the problem discussed here, four free parameters have to be assumed at random or by experience with the problem itself. This means that there are ∞ 4 possible assumptions of initial dimensions. How can the best be found? After having chosen them, the best main dimensions of the cams can be determined exactly when the admissable transmission angle is prescribed. This is the major significance of the paper! As former papers of the author have shown, the transmission angle is determined by the position of the instant centers, which relate to the first derivatives of motion, i.e. the velocity ratio of a function or the tangent of a curve. After a simple example (producing a straight line), a more complicated problem is solved. In a machine, a workpiece is to be transported between three dwell stations in the best way. It is described as follows: The dwells have to be realized exactly, that is, without approximation; (this can easily be fulfilled by having circular arcs on the cams). Then the motion between the dwell stations has to be optimized to give the smallest possible accelerations. As a straight line is still the shortest connection of two points, an equilateral triangle is prescribed as the curve to be produced. In its corners the dwells take place. Between the dwells the cycloidal function is used as one of the best laws of motion.

Conjugate positions for planar four-bar mechanisms and corresponding special forms of the coupler-curves

Four-bar mechanisms and their variations yield two positions for which the output angle β or the coupler angle γ is identical. These positions are called “conjugate positions” and their significance is discussed. The kinematic inversions of mechanisms in certain conjugate positions leads to new mechanisms with special coupler curves. Coupler points which trace a path with two cusps are easily determined. A similar treatment of the non-turnable double rocker yields coupler curves with three cusps. Requirements for symmetrical coupler curves with two or three cusps can be satisfied.

Kinematic Analysis of Spatial Mechanisms by Means of Constant-Distance Equations

The concept and use of constant-distance equations for the kinematic analysis of linkages are presented. The procedure is based on the fact that a constant-distance equation is formulated, wherever the distance between two pair-centers of a rigid link remains constant throughout its motion. This results in a much simpler kinematic analysis of the linkage. To illustrate the procedure and the feasibility of the method, the cases of spatial RRRR– and RGCR-mechanisms with coupler points are considered. The technique is well suited to digital computer analysis of complex mechanisms; extensions to dimensional synthesis as well as to dynamic and mobility analysis are possible.

Synthesis of Crank-Links

Solution procedures, which form part of the synthesis of four-bar linkage coupler point path generators, are shown to be applicable to the simpler problem of synthesizing crank-links with prescribed relationships between the rotation of the crank and the position of the free end of the link.

Higher order analysis of spatial coupler curves

The present paper summarizes developments of numerical matrix methods on the higher order analysis of spatial coupler curves generated by spatial mechanisms. Previously introduced differential displacement matrices and the newly introduced purely geometrical matrices are investigated as “operators” in the matrix operational method. The two methods are compared, and numerical examples for each of the methods are given. Without restriction to any particular spatial mechanism, or any special coupler point, the methods provide a procedure for obtaining the high-order characteristics of coupler curves such as curvature, torsion and spatial center of curvature.

Roberts’ Cognate of Space Five-Link RHHHH and HHRHH Mechanisms

Using the principle of similarly varying triangles, the existence of the coupler cognate of space five-link RHHHH and HHRHH mechanisms is established for the general case in which the coupler point is arbitrarily located in the plane of one of their coupler links.

Coupler Cognate of Geared Five-Link Mechanism

Using Cayley’s construction of parallelograms and similar triangles, the existence of the coupler cognate of the geared five-link mechanism is established for the general case in which the coupler point is arbitrarily located in the plane of one of the coupler links. The Cayley diagram for the geared five-link mechanism is proposed. Its extension leads to the coupler cognates of plane, single-loop, geared, multilink mechanisms.

Coupler cognate mechanisms of certain parallelogram forms of watt's six-link mechanism

Utilizing the principle of stretch rotation and the principle of similarly varying triangle, existence of coupler cognates of Watt's six-link mechanism is investigated. It is shown that for a coupler point located in the plane of the coupler link DE (see Fig. 1) there are two additional stationary pivots, and for the coupler link CD, there exist doubly infinite stationary pivots. These stationary pivots are capable of providing an infinite number of cognate mechanisms.

Coordination of Coupler-Point Positions and Crank Rotations in Connection With Roberts’ Configuration

In this paper a geometrical solution is given to the problem of finding four-bar linkages having 5 given coupler-point positions coordinated with 4 given crank angles. It may also be possible to find solutions if one given coupler-point position together with a given crank angle, corresponding to this position, is replaced by two given link lengths. It has been found thus far that four-bar linkages, satisfying the stated conditions, are easy to obtain if the problem is related to Roberts’ law. To get a view of the possibilities, the degenerations of cognate circle-point curves and cognate center-point curves, linked to each other by the configuration (CR) of Roberts, are investigated. As an example at the end of the paper a straight-line mechanism has been designed so that the coupler point moves along this line with an approximately uniform velocity.