Gordon R. Pennock

The effects of the generating pin size and placement on the curvature and displacement of epitrochoidal gerotors

Abstract Gerotors do not have auxiliary gears in general, therefore the motion of one lobed gear relative to the other lobed gear is produced by the forces between the two contacting lobes. The contact forces, however, result in wear in one or both lobed gears which, in turn, reduces the life of the mechanism. As a gerotor cannot be adjusted to compensate for the wear, it is important that the contact forces be kept to a minimum. It is well known that the wear rate can be reduced by decreasing the curvature (or increasing the radius of curvature) of the lobes. The curvature is a function of the size and the placement of the pins which generate the lobe shape. In this paper, relationships are derived which show the influence of the trochoid ratio, the pin size ratio, and the radius of the generating pin on the curvature of the epitrochoidal gerotor. The relationships provide geometric insight into the design of gerotors which can save time and effort in the manufacturing process. In addition, the results are combined with previously published formulate for the pocket displacement to obtain design charts. These charts can be easily used by the designer to predict the effect of the geometry on the performance of a gerotor. An example is presented, which shows how a commercially available gerotor can be modified to give a 48.7% increase in the minimum radius of curvature with only a 3.2% decrease in the displacement and no change in the overall size of the gerotor.

Geometry for trochoidal-type machines with conjugate envelopes

Abstract This paper presents unified and compact equations describing the geometry and the geometric properties of the different types of trochoid. For the first time, the double-generation theorem is expressed in an explicit manner and a complete classification of all trochoids is documented. Unified and compact equations describing the geometric properties of a conjugate envelope are also presented in the paper. A new type of conjugate envelope for a given trochoid is discovered. An important contribution to the existing literature on trochoidal-type machines is the derivation of closed-form parametric equations for nine types of conjugate envelope. These equations provide significant geometrical insight into the design and analysis of trochoidal-type machines. The paper also presents the necessary and sufficient conditions for a closed type 1 conjugate envelope. Finally, the paper includes a detailed discussion of the characteristics and the relationships of the different types of trochoid and conjugate envelope.

Extension of Graph Theory to the Duality Between Static Systems and Mechanisms

This paper is a study of the duality between the statics of a variety of structures and the kinematics of mechanisms. To provide insight into this duality, two new graph representations are introduced; namely, the flow line graph representation and the potential line graph representation. The paper also discusses the duality that exists between these two representations. Then the duality behveen a static pillar system and a planar linkage is investigated by using the flow line graph representation for the pillar system and the potential line graph representation for the linkage. A compound planetary gear train is shown to be dual to the special case of a statically determinate beam and the duality between a serial robot and a platform-type robot, such as the Stewart platform, is explained. To show that the approach presented here can also be applied to more general robotic manipulators, the paper includes a two-platform robot and the dual spatial linkage. The dual transformation is then used to check the stability of a static system and the stationary, or locked, positions of a linkage. The paper shows that two novel platform systems, comprised of concentric spherical platforms inter-connected by rigid rods, are dual to a spherical six-bar linkage. The dual transformation, as presented in this paper, does not require the formulation and solution of the governing equations of the system under investigation. This is an original contribution to the literature and provides an alternative technique to the synthesis of structures and mechanisms. To simplify the design process, the synthesis problem can be transformed from the given system to the dual system in a straightforward manner.

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.

A Nondegenerate Kinematic Solution of a Seven-Jointed Robot Manipulator

Kinematically simple manipulators with six joints have unavoidable degenerate configurations that correspond to certain positions or orientations of the end-effector within the workspace. In these degenerate configurations, the end-effec tor of the manipulator is not capable of a motion with six independent degrees of freedom. Such a manipulator reduces the flexibility of a manufacturing system because manufac turing processes need to be planned so that positions or ori entations of the end-effector that correspond to degenerate configurations of the manipulator are not required. In this paper a nondegenerate kinematic solution is devel oped for a kinematically simple seven-jointed robot manipu lator. The solution guarantees that the manipulator is capa ble of positioning and orienting its end-effector within the workspace without ever finding itself in a degenerate configu ration. The nondegenerate kinematic solution is in closed form and is developed from well-known methods and results in three-dimensional kinematics. It is believed that this seven-jointed robot manipulator will be most useful in flexible manufacturing systems.

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 utilised 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).

Function Generation With Finitely Separated Precision Points Using Geometric Constraint Programming

This paper extends geometric constraint programming (GCP) to function generation problems involving large numbers of finitely separated precision points and complex mechanisms. In parametric design software, GCP uses the sketching mode to graphically impose geometric constraints in kinematic diagrams and the numerical solvers to solve the relevant nonlinear equations without the user explicitly formulating them. For function generation, the same approach can be applied to any mechanism, requiring no unique algorithms. Implementation is straightforward, so the designer can quickly generate solutions for a large number of precision points and/or with complex mechanisms to accurately match the function. Examples of function generation with a four-bar linkage, a Stephenson III six-bar linkage, and a seven-bar linkage with a mobility of two are presented.

Path curvature of a geared seven-bar mechanism

Abstract 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.

Graphical Methods to Locate the Secondary Instant Centers of Single-Degree-of-Freedom Indeterminate Linkages

This paper presents graphical techniques to locate the unknown instantaneous centers of zero velocity of planar, single-degree-of-freedom, linkages with kinematic indeterminacy. The approach is to convert a single-degree-of-freedom indeterminate linkage into a two-degree-of-freedom linkage. Two methods are presented to perform this conversion. The first method is to remove a binary link and the second method is to replace a single link with a pair of links connected by a revolute joint. First, the paper shows that a secondary instant center of a two-degree-of-freedom linkage must lie on a unique straight line. Then this property is used to locate a secondary instant center of the single-degree-of-freedom indeterminate linkage at the intersection of two lines. The two lines are obtained from a purely graphical procedure. The graphical techniques presented in this paper are illustrated by three examples of single-degree-of-freedom linkages with kinematic indeterminacy. The examples are a ten-bar linkage with only revolute joints, the single flier eight-bar linkage, and a ten-bar linkage with revolute and prismatic joints.

Development of a trajectory generation method for a five-axis NC machine

This paper presents a real-time trajectory generation method and control approach for a five-axis NC machine. The spatial trajectory of the tool of the five-axis machine is described by a ruled surface, and the differential motion parameters of the tool are obtained from the curvature theory of the ruled surface. The controller computes position, orientation, and the differential motion parameters of the tool within a specified sampling period. The Ferguson geometric modeling technique is used to represent the tool trajectory as a ruled surface. The proposed method produces a smoother part surface and requires substantially less machining time when compared to conventional off-line approaches. The real-time control approach, based on the curvature theory of a ruled surface, is believed to be an original contribution to the precise control of a five-axis NC machine.