Ferdinand Freudenstein

Synthesis of kinematic structure of geared kinematic chains and other mechanisms

Abstract Using network concepts and combinatorial analysis, methods have been developed for the classification and enumeration of mechanisms according to kinematic structure. The structural complexity, which can be accommodated, is sufficient to permit the systematic enumeration of many classes of mechanisms. The application of the theory to one such class (gear transmissions and differential drives) is described in detail.

Numerical Solution of Systems of Nonlinear Equations

For sinmltaneous nonlinear equations, the convergence of all functional iteration procedures is dependent upon a good initial approximation to the desired root. In this paper, the restriction on the choice of the initial approximation has been circumvented by dividing each problem into a number of subsidiary problems in accordance with a procedure which is essentially a numerical adaptation of the implicit function theorem. Additional procedures are developed for dealing with sets of equations with singularities in the domain of the iteration. Introduction Newtons method, because of its quadratic convergence, is mathematically the most preferable of the several known methods for the solution of systems of nonlinear equations [1]. Practically, however, a very important limitation (on Newtons method, and in fact, all of the so-called functional iteration methods) is that it does not generally converge to some solution from an arbitrary starting point. Thus Newtons method may fail to converge if the initial estimate is not sufficiently close to the root. The size of the domain of convergence depends upon the system of equations. For real algebraic equations, generally, the size of the domain of convergence is inversely related to the degree and number of equations. Therefore, one finds that for two simultaneous second degree equations almost any initial estimate will lead to one of the roots, while for eight simultaneous tenth degree equations the domain becomes much smaller, and it may be very difficult to obtain an initial estimate from which the iteration converges. The authors and others [2, 3, 4] have found that, even in physically oriented problems, this need for a good initial estimate may become a severe restriction which, in effect, renders the functional iteration techniques useless. This paper describes several algorithms which greatly relax the restrictions on the domain of the initial estimate. Although the discussion is limited to Newtons method, the reader should note that the algorithms are general and can be used in conjunction with any of the functional iteration techniques. )

Kinematics and statics of a coupled epicyclic spur-gear train

Abstract The purpose of this paper is essentially tutorial: to help recast the classical textbook chapter on gear trains in a modern vein. Hopefully this involves general methods, avoiding ad-hoc reasoning and unnecessary complexity. We believe that recently developed methods [1,2,4], employing a graph representation of kinematic structure, are suitable for this purpose. The procedure for kinematic analysis, force analysis and power-flow determination are outlined in simple step fashion, using, for purposes of illustration, a fairly complex, coupled epicyclic drive shown by Glover [5]. It is hoped that the simplicity of the method will commend itself both to the engineering student as well as to the practicing engineer.

Application of Line geometry to theoretical kinematics and the kinematic analysis of mechanical systems

Abstract The theory of screws as developed by R. S. Ball, is concerned principally with rigid-body motions in three-dimensional Euclidean space. A number of scattered references, notably F. Klein [18], have mentioned algebraic approaches to the subject. In this paper which is essentially tutorial, we have used concepts of line geometry in order to present such a self-contained algebraic formulation. Other approaches to the subject, which are not considered here, are possible and useful: for example, motor algebra (F. M. Dimentberg) and surface geometry (K. H. Hunt, K. J. Waldron). The algebraic treatment is both general and well-adapted to kinematic analysis and numerical methods. These applications have been described.

The basic concepts of Polya's theory of enumeration, with application to the structural classification of mechanisms

Abstract The purpose of these notes is essentially tutorial: to present to engineers the basic concepts of Polyas theory in a self-contained manner. Following a review of preliminary concepts (permutations, groups and graphs), Polyas “Hauptsatz” is stated without proof and its use illustrated with reference to the structural classification of mechanisms.

Transmission optimization of spatial 4-link mechanisms

Abstract The design of plane crank-and-rocker linkages with given rocker swing angle, corresponding crank rotation and optimum force transmission are well known problems in kinematic synthesis. This investigation is concerned with the transmission optimization of the skew crank-and-rocker linkage and the skew slider-crank mechanism. The algebraic loci of the moving joints are used to express the lengths of the moving links as a function of a single parameter, the optimum value of which is obtainable as a constrained minimization problem involving 1-parameter scanning. The results are presented in tables as well as in design charts.

An application of graph theory and nonlinear programming to the kinematic synthesis of mechanisms

Abstract In this investigation a systematic “start-from-zero” approach for the design of mechanisms has been described and applied to the creation of an automotive windshield-wiper mechanism. Following specification of the design goals and restrictions, the potentially useful kinematic structures have been generated with the aid of graph theory. A preliminary functional screening of these structures has yielded several potentially useful mechanisms. The type synthesis was followed by dimensional synthesis, formulated as a nonlinear programming problem. Physical design goals and constraints were transformed into the objective and penalty functions and then optimized. The final result was a proportioned potentially optimal mechanism. An alternative approach to the dimensional synthesis of the generated kinematic structures in terms of precision points has been discussed as well.

Optimization of the workspace of a three-link turning-pair connected robot arm

The workspace of a three-axis, turning-pair connected robot arm has been optimized using algebraic criteria for extreme axial and radial reach and the elimination of voids (Freuden stein and Primrose 1984). Based on these criteria, an effi cient, systematic search procedure has been developed for maximizing the workspace and the rvorkspace-to-void ratio of the R3 robot arm.

Synthesis of two-degree-of-freedom linkages —A feasibility study of numerical methods of synthesis of bivariate function generators

Abstract An algebraic formulation is presented for the dimensional synthesis of plane seven-link two-degree-of-freedom mechanisms; a numerical method for the solution of the equations of synthesis has been programmed and evaluated; the scope and feasibility of present and future developments in this area have been assessed.

Design of geared 5-bar mechanisms for unlimited crank rotations and optimum transmission

Abstract This investigation is concerned with the determination of the rotatability of the geared cranks and the optimization of transmission characteristics of geared 5-bar linkages. Algebraic solutions are shown to be feasible only in a few, special cases. An efficient computer-aided procedure, however, can be used for the general case and for the development of design charts. These charts, which are given for the mechanisms with gear ratio minus one, can be used to determine whether a mechanism of given proportions possesses unlimited crank rotations and to select the floating-link dimensions yielding optimum transmission.