Aronhold-Kennedy Theorem Research Articles

A Screw Theory Approach to Computing the Instantaneous Rotation Centers of Indeterminate Planar Linkages

This paper presents a screw theory approach for the computation of the instantaneous rotation centers of indeterminate planar linkages. Since the end of the 19th century, the determination of the instantaneous rotation, or velocity centers of planar mechanisms has been an important topic in kinematics that has led to the well-known Aronhold–Kennedy theorem. At the beginning of the 20th century, it was found that there were planar mechanisms for which the application of the Aronhold–Kennedy theorem was unable to find all the instantaneous rotation centers (IRCs). These mechanisms were denominated complex or indeterminate. The beginning of this century saw a renewed interest in complex or indeterminate planar mechanisms. In this contribution, a new and simpler screw theory approach for the determination of indeterminate rotation centers of planar linkages is presented. The new approach provides a simpler method for setting up the equations. Furthermore, the algebraic equations to be solved are simpler than the ones published to date. The method is based on the systematic application of screw theory, isomorphic to the Lie algebra, se(3), of the Euclidean group, SE(3), and the invariant symmetric bilinear forms defined on se(3).

Instant Center Identification of Single-Loop Multi-DOF Planar Linkage Using Virtual Link

Instant center is an important kinematic characteristic which can be used for velocity and singularity analysis, configuration synthesis and dynamics modeling of multi-degree of freedom (multi-DOF) planar linkage. The Aronhold–Kennedy theorem is famous for locating instant centers of four-bar planar linkage, but for single-loop multi-DOF linkages, it fails. Increasing with the number of the links of single-loop multi-DOF planar linkages, the lack of link relationship makes the identification of instant center become a recognized difficulty. This paper proposes a virtual link method to identify instant centers of single-loop multi-DOF planar linkage. First, three types of instant centers are redefined and the instant center identification process graph is introduced. Then, based on coupled loop chain characteristic and definition of instant center, two criteria are presented to convert single-loop multi-DOF planar linkage into a two-loop virtual linkage by adding the virtual links. Subsequently, the unchanged instant centers are identified in the virtual linkage and used to acquire all the instant centers of original single-loop multi-DOF planar linkage. As a result, the instant centers of single-loop five-bar, six-bar planar linkage with several prismatic joints are systematically researched for the first time. Finally, the validity of the proposed method is demonstrated using loop equations. It is a graphical and straightforward method and the application is wide up to single-loop multi-DOF N-bar (N ≥ 5) planar linkage.

Replacement in a Plane Geared Linkage of High Kinematic Pairs with Lower Pairs

The analysis of constraints in plane mechanisms is an urgent problem in the theory of machines and mechanisms. Although kinematic pairs’ classification has been known for a long time, the issue of the conjugation of links, being at the heart of the analysis and synthesis of mechanisms and machines, is of considerable theoretical and practical interest and continues to attract scientists. One of the tasks that are solved in the process of analysis and synthesis of the structures of mechanisms is the re-placement of higher kinematic pairs by lower ones. As a rule, such a replacement is made to identify kinematic chains of zero mobility, Assur's structural groups, in a mechanism. The replacement may also aim at obtaining the necessary kinematic relations. That is because specific computational difficulties hamper the kinematic analysis of chains with higher kinematic pairs due to the relative sliding and shape irregularity of mating surfaces. Yet, the use of replacements to obtain kinematic and transmission functions is difficult due to nonisomorphism of the equivalent mechanism. Simultaneously, for mixed-type mechanisms, which include geared linkages, the equivalent replacement will allow unifying the kinematic analysis methods. The paper suggests the technology of replacing higher kinematic pairs with links with lower pairs as applied to a plane geared linkage. The technology is based on the properties of the involute of a circumference. The paper proved the structural and kinematic equivalence of such a replacement. The isomorphism of the equivalent linkage will enhance the kinematic analysis, make it possible using kinematic functions, and applying methods based on the instantaneous relative rotations of links, in particular, the Aronhold-Kennedy theorem. Another application of the replacement method presented in the paper will be the expansion of opportunities for identifying idle constraints in the mechanism.

A screw theory approach to compute instantaneous rotation axes of indeterminate spherical linkages

This paper presents a screw theory approach for the computation of the instantaneous rotation axes of indeterminate spherical linkages. Since the last part of the XIX century, the determination of the instantaneous rotation, or velocity, centers of planar mechanisms has been studied using the Aronhold-Kennedy theorem, extended to the spatial case until the second part of last century. In the beginning of the XX century, it was found that there were planar mechanisms for which the application of the Aronhold-Kennedy theorem was unable to find all the instantaneous rotation centers; these mechanisms where denominated complex or indeterminate. The beginning of this century saw a renewed interest on the complex or indeterminate mechanisms, both planar and spherical; and very soon, taken advantage of the results obtained for planar mechanisms, there were several methods for the determination of the indeterminate instantaneous rotation axes. The new screw theory approach, presented here, provides a more straightforward method for setting up the equations; furthermore, the algebraic equations to be solved are simpler than the ones published up to date. In addition, the method is part of a comprehensive method for the determination of the instantaneous screw axes of spatial mechanisms or their many special cases. The method is based on a systematic application of screw theory, which is isomorphic to the Lie algebra, se(3), of the Euclidean group, SE(3), and the invariant symmetric bilinear forms defined on se(3).

Extended Camus Theory and Higher Order Conjugated Curves

According to Camus’ theorem, for a single degree-of-freedom (DOF) three-body system with the three instant centers staying coincident, a point embedded on a body traces a pair of conjugated curves on the other two bodies. This paper discusses a fundamental issue not addressed in Camus’ theorem in the context of higher order curvature theory. Following the Aronhold–Kennedy theorem, in a single degree-of-freedom three-body system, the three instant centers must lie on a straight line. This paper proposes that if the line of the three instant centers is stationary (i.e., slide along itself) on the line of the instant centers, a point embedded on a body traces a pair of conjugated curves on the other two bodies. Another case is that if the line of the three instant centers rotates about a stationary point, the stationary point embedded on a body also traces a pair of conjugated curves on the other two bodies. The paper demonstrates the use of instantaneous invariants to synthesize such a three-body system leading to a conjugate curve-pair generation. It is a supplement or extension of Camus’ theorem. Camus’ theorem may be regarded as a special singular case, in which all three instant centers are coincident.

A proposal for a formula of absolute pole velocities between relative poles

In this paper is proposed a vectorial equation that relates the absolute pole velocities of three moving rigid bodies with a planar motion of general type. From this equation, it is possible to obtain a relation between the pole velocities of the three mathematical points, related between them by the Aronhold-Kennedy Theorem. The formula allows the calculation of one of the pole velocities from the other two, being known the angular velocities and accelerations of the moving bodies. It is applicable regardless of whether the instantaneous centers (poles) are located on physical points on the linkage or not. Illustrative examples of the application of the formula on representative planar linkages are included. In the final section, is discussed a similar concept associating a mathematical point to the curvature centers of a point's path, so called centroma.

A survey of instantaneous poles for a class of two-degree-of-freedom spherical mechanisms

This paper is a detailed exploration of instantaneous poles for a class of two-degree-of-freedom (two-DOF) spherical mechanisms (SMs) with seven links or bars. For two-DOF SMs, the secondary instantaneous poles (the ones which cannot be found by direct inspection) must lie on the specified great circles. For many of these mechanisms, however, some of these great circles cannot be obtained by a direct application of Aronhold-Kennedy theorem. This paper presents geometrical and analytical techniques to locate these unknown great circles for three topologies of seven-bar two-DOF SMs.

GEOMETRICAL METHODS TO LOCATE SECONDARY INSTANTANEOUS POLES OF SINGLE-DOF INDETERMINATE SPHERICAL MECHANISMS

A single-degree-of-freedom (DOF) indeterminate spherical mechanism is defined as a mechanism for which it is not possible to find all the instantaneous poles by direct application of the Aronhold-Kennedy theorem. This paper shows that a secondary instantaneous pole of a two DOFs spherical mechanism lies on a unique great circle instantaneously. Using this property, two geometric methods are presented to locate secondary instantaneous poles of indeterminate single DOF spherical mechanisms. Common approach of the methods is to convert a single DOF indeterminate spherical mechanism into a two DOFs mechanism and then to find two great circles that the unknown instantaneous pole lies on the point of intersection of them. The presented methods are directly deduced from a work done for indeterminate single DOF planar mechanisms.DOI: http://dx.doi.org/10.3329/jme.v41i2.7471

A study of the instantaneous centers of velocity for two-degree-of-freedom planar linkages

Abstract The secondary instantaneous centers of velocity for two-degree-of-freedom planar linkages must lie on straight lines. For many of these linkages, however, some of these lines cannot be obtained by a direct application of the Aronhold–Kennedy theorem. This paper, therefore, will present both graphical and analytical techniques to locate these unknown lines of centers for certain types of two-degree-of-freedom linkages. First the techniques are applied to the three topologies of the seven-bar linkage, two of which contain a four-bar chain, and the unknown lines of centers are obtained for each topology. Then the techniques are applied to the 35 topologies of the nine-bar linkage. The paper includes a discussion of these topologies and presents three examples for illustrative purposes. The techniques can locate the unknown lines of centers for all, but one, of the 35 topologies. The techniques presented in this paper provide geometric insight into the first-order instantaneous kinematics of two-degree-of-freedom planar linkages.

A GRAPHICAL-ANALYTICAL TECHNIQUE FOR THE SYNTHESIS OF NON-CIRCULAR GEARS IN PATH-GENERATING GEARED FIVE-BAR MECHANISMS

The paper presents a graphical-analytical technique for the synthesis of non-circular gears in path-generating geared five-bar mechanisms. In such mechanisms, a two degree-of-freedom (dof) five-bar linkage is integrated by a pair of non-circular gears to precisely guide a coupler point along a prescribed planar trajectory. The synthesis method proposed here considers the most general case, where the prescribed trajectory consists of a series of open curves. Each segment of the prescribed path identifies a phase of the mechanism motion that is referred to as requested motion branch. In each of these phases, for any prescribed position of the coupler point, the inverse kinematic analysis of the linkage and the Aronhold-Kennedy theorem are used to identify the actual configuration of the system and locate the instantaneous centre of the relative motion between the two cranks of the linkage. The regions of the gear’s centrodes, corresponding to the requested motion branches, are thus synthesized. These regions are connected to each other by using proper polynomial functions, as to guarantee a continuous and cyclic motion of the mechanism. An example is illustrated where the requested coupler point trajectory consists of a series of straight line segments.

Study of Dead-Centre Positions of Single-Degree-of-Freedom Planar Linkages Using Assur Kinematic Chains

The article presents an original technique, using the concept of Assur kinematic chains (AKCs), to determine whether a single-degree-of-freedom planar linkage is in a dead-centre position, i.e. a position where the input link is instantaneously stationary. An AKC is a special structure with mobility zero from which it is not possible to obtain a simpler substructure of the same mobility by removing one or more links. The article presents the concept of modularization of planar linkages into AKC based on the choice of the input link. Then, the article presents the constraints on the locations of the instantaneous centres of zero velocity (or instant centres) for a single-degree-of-freedom planar linkage to be in a stationary configuration, i.e. a configuration where one, or more, of the links is instantaneously stationary. The article shows that constraints on the locations of the instant centres for a stationary configuration are satisfied if an AKC, as part of the linkage, gains a degree of freedom. As the modularization of a planar linkage is based on the choice of the input link, the stationary configurations, determined by this method, are in fact dead-centre positions. Finally, this method is applied to indeterminate linkages, i.e. a class of single-degree-of-freedom planar linkages for which it is not possible to locate all the secondary (or unknown) instant centres by the direct application of the Aronhold—Kennedy theorem.

A Study of the Duality Between Planar Kinematics and Statics

This paper provides geometric insight into the correlation between basic concepts underlying the kinematics of planar mechanisms and the statics of simple trusses. The implication of this correlation, referred to here as duality, is that the science of kinematics can be utilized in a systematic manner to yield insight into statics, and vice versa. The paper begins by introducing a unique line, referred to as the equimomental line, which exists for two arbitrary coplanar forces. This line, where the moments caused by the two forces at each point on the line are equal, is used to define the direction of a face force which is a force variable acting in a face of a truss. The dual concept of an equimomental line in kinematics is the instantaneous center of zero velocity (or instant center) and the paper presents two theorems based on the duality between equimomental lines and instant centers. The first theorem, referred to as the equimomental line theorem, states that the three equimomental lines defined by three coplanar forces must intersect at a unique point. The second theorem states that the equimomental line for two coplanar forces acting on a truss, with two degrees of indeterminacy, must pass through a unique point. The paper then presents the dual Kennedy theorem for statics which is analogous to the well-known Aronhold-Kennedy theorem in kinematics. This theorem is believed to be an original contribution and provides a general perspective of the importance of the duality between the kinematics of mechanisms and the statics of trusses. Finally, the paper presents examples to demonstrate how this duality provides geometric insight into a simple truss and a planar linkage. The concepts are used to identify special configurations where the truss is not stable and where the linkage loses mobility (i.e., dead-center positions).

Curvature theory for the double flier eight-bar linkage

This paper presents a graphical technique to obtain the radius of curvature of the path traced by a coupler point of a planar, single-degree-of-freedom, indeterminate eight-bar linkage commonly referred to as the double flier linkage. The first step is to find the pole for the instantaneous motion of the coupler link; i.e., the point coincident with the absolute instant center of the coupler link. Since the double flier is a linkage with kinematic indeterminacy then the Aronhold–Kennedy theorem cannot locate this instant center. The paper, therefore, begins with a novel technique, which requires few geometric constructions, to locate this instant center. Then the paper focuses on the graphical technique to determine the radius of curvature of the path of an arbitrary coupler point for a given position of the input link. The technique begins by obtaining an equivalent five-bar linkage and four kinematic inversions of this linkage. A four-bar linkage is obtained from each inversion. Finally, the systematic procedure provides a four-bar linkage in which the motion of the coupler link is equivalent up to, and including, the second order properties of motion of the coupler of the original double flier linkage. The radius of curvature of the path traced by the coupler point is then obtained from the well-known Euler–Savary equation.

Path Curvature of the Single Flier Eight-Bar Linkage

This paper presents a graphical technique to locate the center of curvature of the path traced by an arbitrary coupler point of the single flier eight-bar linkage. The first step is to locate the pole for the instantaneous motion of the coupler link; i.e., the point in the fixed plane coincident with the absolute instant center of the coupler link. Since the single flier is an indeterminate linkage, comprised of one four-bar and two five-bar chains, then the Aronhold-Kennedy theorem cannot locate this instant center. The paper presents a novel graphical technique which can locate this instant center in a direct manner. Then the paper focuses on a graphical method to locate the center of curvature of the path traced by the coupler point. The method locates six equivalent four-bar linkages for the two five-bar chains, investigates six kinematic inversions and obtains a four-bar linkage from each inversion. This systematic procedure produces a four-bar linkage with a coupler link whose motion is equivalent up to, and including, the second-order properties of motion of the single flier coupler link. The radius of curvature and the center of curvature of the path traced by the coupler point can then be obtained in a straightforward manner from the Euler-Savary equation.

Path curvature of a geared seven-bar mechanism

This paper presents a graphical technique to locate the center of curvature of the path traced by a coupler point of a planar, single-degree-of-freedom, geared seven-bar mechanism. Since this is an indeterminate mechanism then the pole for the instantaneous motion of the coupler link; i.e., the point coincident with the instantaneous center of zero velocity for this link, cannot be obtained from the Aronhold–Kennedy theorem. The graphical technique that is presented in the first part of the paper to locate the pole is believed to be an important contribution to the kinematics literature. The paper then focuses on the graphical technique to locate the center of curvature of the path traced by an arbitrary coupler point. The technique begins with replacing the seven-bar mechanism by a constrained five-bar linkage whose links are kinematically equivalent to the second-order properties of motion. Then three kinematic inversions are investigated and a four-bar linkage is obtained from each inversion. The motion of the coupler link of the final four-bar linkage is equivalent up to and including the second-order properties of motion of the coupler of the geared seven-bar. Then the center of curvature of the path traced by an arbitrary coupler point can be obtained from existing techniques, such as the Euler–Savary equation. An analytical method, referred to as the method of kinematic coefficients, is presented as an independent check of the graphical technique.

A Graphical Method to Find the Secondary Instantaneous Centers of Zero Velocity for the Double Butterfly Linkage

The Aronhold-Kennedy theorem cannot locate all of the instantaneous centers of zero velocity for a planar, single-degree-of-freedom, indeterminate linkage. This paper presents a graphical technique that will locate the secondary instantaneous centers of zero velocity for a well-known indeterminate linkage; namely, the double butterfly linkage. Only one of the secondary instant centers of this eight-bar linkage needs to be located with the proposed technique; the remaining instant centers can then be located using the Aronhold-Kennedy theorem. The first step in the graphical method is to regard the double butterfly linkage as two six-bar linkages. This is accomplished by replacing the ternary link pinned to the ground by two binary links, removing the pin which connects the two coupler links, and attaching a slider to each coupler link at the coupler pin. The second step is to reduce two of the five-bar loops of the double butterfly linkage to four-bar loops by instantaneously freezing the aforementioned binary links. The paper shows how these two important steps are used to locate the absolute instant centers for the two coupler links of this indeterminate eight-bar linkage.

The Design of a Novel Pure-Rolling Transmission to Convert Rotational into Translational Motion

The Design of a Novel Pure-Rolling Transmission to Convert Rotational into Translational Motion