2.1 Graphical Position Analysis of Mechanisms

2.1 Graphical Position Analysis of Mechanisms#

Graphical position analysis is a visual and intuitive exercise that allows for a fundamental understanding of the mechanism’s motion.

Principles and Tools#

It is a “truly trivial” method that can be performed “with only high-school geometry.” The basic tools required are a ruler, a compass, and a protractor.

General Procedure#

  • For a mechanism with one degree of freedom (DOF), such as a four-bar linkage, it is sufficient to define one parameter (usually the angle of the input link, \(q_2\)) to establish the positions of all other links.

  • Draw the ground link (link 1) and the input link (link 2) to a convenient scale, positioning link 2 at its input angle \(q_2\).

  • With the compass, draw an arc from the end of link 2 (point A) with a radius equal to the length of link 3. Simultaneously, draw a second arc from the fixed pivot of link 4 (point O4) with a radius equal to the length of link 4.

  • The two intersections of these arcs (points B and B’) define the two possible solutions to the position problem: the “open” and “crossed” circuit configurations. The configuration (open or crossed) depends solely on how the links are assembled.

Advantages and Limitations#

  • Visual Insight: Offers excellent visual feedback that helps build an intuitive understanding of the underlying kinematic principles and potential motion issues.

  • Quick Verification: Provides a fast method to obtain a solution or to verify a result from a computer program.

  • Precision: When performed with CAD or dynamic geometry software (e.g., AutoCAD, GeoGebra), the method is highly precise. The limitation of precision only applies when the construction is done manually with physical drawing tools.

  • Repetitiveness: The process can become tedious if many positions need to be analyzed, as each new input position requires a new construction, although software can streamline this.

Applicable Mechanisms#

The graphical method is versatile and can be applied to a wide variety of planar linkages.

  • Single-DOF Linkages: It works perfectly for simple linkages like the four-bar and slider-crank mechanisms. It is also effective for more complex single-degree-of-freedom mechanisms, such as the Watt six-bar linkage. However, some topologies, like the Stephenson II six-bar linkage, cannot be solved with the direct compass-arc intersection method and require more advanced geometric constructions.

  • Multi-DOF Linkages: The method is also easily applied to mechanisms with multiple degrees of freedom. In such cases, a position for each input link must be defined to determine the position of the rest of the mechanism.

Geogebra Integration#

For this section, integrating Geogebra iframes will be invaluable. It allows readers to interact directly with the mechanisms, varying the input angle and observing how the link positions change in real-time.

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