A Proposed Method to Group the Solutions From Dimensional Synthesis: Planar Triads for Six Precision Position

In this paper, a method has been proposed to group into six sets the infinite number of solutions from dimensional synthesis of planar triads for six precision positions. The proposed method reveals the relationships between the different configurations of the compatibility linkage and the sets of numerical solutions from dimensional synthesis. By checking the determinant signs and the contunities of values of the sub-Jacobian matrices and their derivatives with respect to the independent angular displacement for all constraint sets in the compatibility linkage, it enables the computer to identify and group the synthesized solutions. Numerical examples have been given to verify the applicability of this method. Six sets of the partial triad Burmester curves have been plotted based on grouped solutions. Suitable solutions can be easily found from the partial triad Burmester curves and utilized for the prescribed design task. This method provides a useful tool to group the dimensional synthesis solutions and enhances the computer automation in the design of linkage mechanisms.

On the Design of Planar and Spherical Pure-Rolling Indexing Cam Mechanisms

A new design of indexing cam mechanisms for parallel and intersecting shafts is presented here in a unified framework, so that both pure rolling and positive motion are achieved. Power losses due to Coulomb friction are eliminated, while producing motions free of jerk discontinuities. The pressure angle is anlyzed and applied to define the positive motion.

Optimal Synthesis of Four-Bar Linkages for Four-Position Rigid-Body Guidance With Selective Tolerance Specifications

In practical rigid-body guidance problems, very often one or more of the design positions need not be generated exactly. Further, extreme accuracy at the design positions is somewhat pointless considering the inherent limitations in linkage manufacturing processes. This emphasizes the requirement of synthesis techniques to be able to handle tolerance specifications on the nominal design positions. A favorable offshoot of the tolerance incorporation would be the accompanying increase in the solution space of the synthesis problem thereby yielding better linkage solutions.

Complete Solutions to Synthesis of Six-Link, Slider-Crank and Four-Link Mechanisms for Function, Path and Motion Generation Using Homotopy With M-Homogenization

This paper presents complete solutions to the function, motion and path generation problems of Watt’s and Stephenson six-link, slider-crank and four-link mechanisms using homotopy methods with m-homogenization. It is shown that using the matrix method for synthesis, applying m-homogeneous group theory, and by defining compatibility equations in addition to the synthesis equations, the number of homotopy paths to be tracked can be drastically reduced. For Watt’s six-link function generators with 6 thru 11 precision positions, the number of homotopy paths to be tracked in obtaining all possible solutions range from 640 to 55,050,240. For Stephenson-II and -III mechanisms these numbers vary from 640 to 412,876,800. For 6, 7 and 8 point slider-crank path generation problems, the number of paths to be tracked are 320, 3840 and 17,920, respectively, whereas for four-link path generators with 6 thru 8 positions these numbers range from 640 to 71,680. It is also shown that for body guidance problems of slider-crank and four-link mechanisms, the number of homotopy paths to be tracked is exactly same as the maximum number of possible solutions given by the Burmester-Ball theories. Numerical results of synthesis of slider-crank path generators for 8 precision positions and six-link Watt and Stephenson-III function generators for 9 prescribed positions are also presented.

On the Dynamic Analysis of Roller Chain Drives: Part II — Case Study

Part I of this paper (5) summarized the previous work and has described the theoretical and computational aspects of a computer-aided procedure which has been developed by the authors for the dynamic analysis of roller chain drives. Lagrange’s equations of motion have been derived by assuming the roller chain to behave as a series of masses lumped at the roller centers and connected by bars of constant axial stiffness. The equations of motion are solved in the time domain until steady state conditions are achieved.

Constrained Optimization of Multi-Degree-of-Freedom Mechanisms for Near-Time Optimal Motions

This paper presents a design method of multi-degree-of-freedom mechanisms for near-time optimal motions. The design objective is to select system parameters, such as link lengths and actuator sizes, so as to minimize the optimal motion time of the mechanism along a given path. The exact time optimization problem is approximated by a simpler procedure that maximizes the acceleration near the end points. Representing the directions of maximum acceleration with the acceleration lines, and the reachability constraints as explicit functions of the design parameters, we transform the constrained optimization to a simpler curve fitting problem that can be formulated analytically. This allows the use of efficient gradient type optimizations, instead of the pattern search optimization that is otherwise required. Examples for optimizing the dimensions of a five-bar planar mechanism demonstrate close correlation of the approximate with the exact solutions, and an order of magnitude better computational efficiency than the previously developed unconstrained optimization methods.

An Automated Inversion Apporach to Closed-Form Kinematic Analysis of Planar Mechanisms

In this paper an inversion approach is developed for the analysis of planar mechanisms using closed-form equations. The vector loop equation approach is used, and the occurrence matrices of the variables in the position equations are obtained. After the loop equations are formed, dependency checking of the unknowns is performed to determine if it is possible to solve for any two equations in two unknowns. For the cases where the closed-form solutions cannot be implemented directly, possible inversions of the mechanism are studied. If the vector loop equations for an inversion can be solved in closed-form, they are identified and solved, and the solutions are transformed back to the original linkage. The method developed in this paper eliminates the uncertainties involved, and the large number of computations required in solving the equations by iterative methods.

A Dynamic Implementation of Keller’s Sketching Rules for Burmester Curves an Approach to Sensitivity Analysis

Keller’s sketching rules for Burmester curves are implemented on the computer for automatic generation of valid and invalid regions for the existence of the curves. The rules are based on the poles of the coupler motion, which in turn are directly related to the precision positions. All six possible unique arrangements of poles are used in the sketching rules. The sketching rules provide a useful heuristic and geometric approach to relating the sensitivity of the curves to small changes in the pole locations. The regions and the curves are dynamically updated as the poles of the specified motion are moved by the user. The new curve obtained is checked to verify satisfaction of original tolerance specification on the precision positions. The procedure shows promise for artificially intelligent approaches to linkage design.

A Dyad Search Algorithm for Solving Planar Linkages Using the Dyad Method

Based on graph representation of planar linkages, a new algorithm was developed to identify the different dyads of a mechanism. A dyad or class II group, is composed of two binary links connected by either a revolute (1) or a slider (0) pair with provision for attachment to other links by lower pair connectors located at the end of each link. There are five types of dyads: the D111, D101, D011, D001, and D010. The dyad analysis of a mechanism is predicated on the ability to construct the system from one or more of the five binary structure groups or class II groups. If the mechanism is complicated and several dyads are involved, the task of identifying these dyads by inspection could be difficult and time consuming for the user. This algorithm allows a complete automation of this task. This algorithm is based on the Dijkstra’s algorithm, for finding the shortest path in a graph, and it is used to develop a computer program, called KAMEL: Kinematic Analysis of MEchanical Linkages, and implemented on an IBM-PC PS/2 model 80. When compared to algorithmic methods, like the Newton-Raphson, the dyad method proved to be a very efficient one and requires as little as one tenth of the time needed by the method using Newton-Raphson algorithm. Moreover, the dyad method yields the exact solution of the position analysis and no initial estimates are needed to start the analysis. This method is also insensitive to the value of the step-size crank rotation, therefore, allowing a very accurate and fast solution of the mechanism at any position of the input link.

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. The circuit characteristics of Stephenson six-bar linkages can be established by examining the interaction of the four-bar and five-bar chains within the linkage. Preliminary synthesis of the four-bar chain introduces specific conditions that must be met by the five-bar chain for all of the precision positions to lie on a single circuit. These conditions are based on the relationship between the four-bar coupler curve and the dyads/triads that are generated from the centerpoint curve of the final synthesis step. The travel of the coupler curve between the precision positions is compared with the reach of the final dyad/triad to establish circuit conditions. The circuit conditions are applied to points along the centerpoint curve of the final synthesis step to identify those curve segments which will produce a mechanism with all precision positions on the same circuit.