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Research Papers

Complete Solution to the Eight-Point Path Generation of Slider-Crank Four-Bar Linkages

[+] Author and Article Information
Hafez Tari1

Virtual Reality and Mechanisms Laboratory, Department of Mechanical Engineering, The University of Maryland, Baltimore County, Baltimore, MD 21250hafez.tari@umbc.edu

Hai-Jun Su

Virtual Reality and Mechanisms Laboratory, Department of Mechanical Engineering, The University of Maryland, Baltimore County, Baltimore, MD 21250haijun@umbc.edu

1

Corresponding author.

J. Mech. Des 132(8), 081003 (Jul 21, 2010) (7 pages) doi:10.1115/1.4001878 History: Received November 01, 2009; Revised May 20, 2010; Published July 21, 2010; Online July 21, 2010

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.

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Copyright © 2010 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Slider-crank four-bar cognates

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Figure 2

(a) A schematic view of a slider-crank four-bar and the vector representation of its geometry along with a typical coupler curve; (b) the displaced configuration

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Figure 3

The solution process using the homotopy continuation method

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Figure 4

A schematic view of a slider-crank four-bar and its coupler curve based on the values given in Table 2

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Figure 5

A slider-rocker four-bar and its coupler curve as a solution to example 1

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Figure 6

Graph of f(x,y) and the sample points on it for example 2

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Figure 7

Six useful solutions for example 2

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