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We study the synthesis of a slider-crank four-bar linkage whose coupler point traces a set of predefined task points. We report that there are at most 558 slider-crank four-bars in cognate pairs passing through any eight specified task points. The problem is formulated for up to eight precision points in polynomial equations. Classical elimination methods are used to reduce the formulation to a system of seven sixth-degree polynomials. A constrained homotopy technique is employed to eliminate degenerate solutions, mapping them to solutions at infinity of the augmented system, which avoids tedious post-processing. To obtain solutions to the augmented system, we propose a process based on the classical homotopy and secant homotopy methods. Two numerical examples are provided to verify the formulation and solution process. In the second example, we obtain six slider-crank linkages without a branch or an order defect, a result partially attributed to choosing design points on a fourth-degree polynomial curve. [DOI: 10.1115/1.4001878]

This paper extends previous work concerning convex reformulations of counterweight balancing by developing a general and numerically efficient design framework for counterweight balancing of arbitrarily complex planar linkages. At the numerical core of the framework is an iterative procedure, in which successively solving three convex optimization problems yields practical counterweight shapes in typically less than 1 CPU s. Several types of counterweights can be handled. The iterative procedure allows minimizing and/or constraining shaking force, shaking moment, driving torque, and bearing forces. Numerical experiments demonstrate the numerical superiority (in terms of computation time and balancing result) of the presented framework compared to existing approaches. [DOI: 10.1115/1.4000532]

A systematic methodology for the design of a statically balanced, single degree-of-freedom planar linkage is presented. This design methodology is based on the concept of conservation of potential energy, formulated by the use of complex number notations as link vectors of the linkage. By incorporating the loop closure equations and the kinematic constraints, the gravitational potential energy of the system can be formulated as the function of the vectors of all ground-adjacent links. The balance of the gravitational potential energy of the system is then accomplished by the elastic potential energy of a zero free-length spring on each ground-adjacent link of the linkage. As a result, spring constants and installation configurations of the ground-attached springs are obtained. Since the variation in the gravitational potential energy of the linkage at all configurations can be fully compensated by that of the elastic potential energy of the ground-attached springs, this methodology provides an exact solution for the design of a general spring balancing mechanism without auxiliary parallel links. Illustrations of the methodology are successfully demonstrated by the spring balancing designs of a general Stephenson-III type six-bar linkage and a Watt-I type six-bar linkage with parallel motion. [DOI: 10.1115/1.3087548] Keywords: static balance, zero free-length spring, planar linkages, potential energy

This paper presents the concept and application of input joint rotation space of linkages and offers updates on the N-bar rotatability laws. A thorough discussion on the joint rotation space of single-loop planar five-bar linkages is first presented. The concept is then extended to spherical linkages and the generalization to N-bar linkages is discussed. It offers a visualization tool for the input joint rotatability and fills up a void in the N-bar rotatability laws regarding the coordination among multiple inputs. It explains the formation of branches and how to establish a one-to-one correspondence between the inputs and the linkage configurations. The applications to multiloop linkages and spatial linkages are highlighted with Stephenson six-bar linkages, geared linkages, and spatial RCRCR mechanisms. These examples exhibit simplicity and benefits of the proposed concept to the mobility analysis of diversified mechanisms. The concept of virtual loop in spatial linkages is proposed and demonstrated with simple RCRCR and Stephenson sixbar mechanisms. [DOI: 10.1115/1.2943299]

Linkage and cam-follower mechanism design are often based on the simplifying assumption of constant drive speed. Usually, however, the mutual interaction between the mechanism and its actuator gives rise to a fluctuating drive speed, possibly resulting in a variety of dynamic problems. This paper provides machine designers and researchers with a structured design approach toward solving such problems. As such, it constitutes both an extension and a complement to the 1979 input torque balancing survey by Berkof. [DOI: 10.1115/1.2920473]