Research Papers

The Spherical Equivalent of Bresse's Circles: The Case of Crossed Double-Crank Linkages

[+] Author and Article Information
Giorgio Figliolini

Department of Civil and Mechanical Engineering,
University of Cassino and Southern Lazio,
Via G. Di Biasio 43,
Cassino (Fr) 03043, Italy
e-mail: figliolini@unicas.it

Jorge Angeles

Department of Mechanical Engineering & CIM,
McGill University,
817 Sherbrooke Street W.,
Montréal, QC H3A 2K6, Canada
e-mail: angeles@cim.mcgill.ca

Manuscript received March 5, 2016; final manuscript received December 9, 2016; published online January 12, 2017. Assoc. Editor: Jian S. Dai.

J. Mechanisms Robotics 9(1), 011014 (Jan 12, 2017) (11 pages) Paper No: JMR-16-1056; doi: 10.1115/1.4035504 History: Received March 05, 2016; Revised December 09, 2016

The subject of Bresse's circles is classical in the kinematics of planar mechanisms. These are the loci of the coupler points that exhibit either zero normal or zero tangential acceleration. Described in this paper is the construction of the spherical equivalent of Bresse's circles, which take the form of an inflexion spherical cubic and a Thales ellipse, respectively. An algorithm is developed to produce these loci for the case of the spherical antiparallelogram. As a byproduct, the corresponding polodes and their evolutes are obtained.

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Fig. 3

Kinematic sketch of the spherical four-bar linkage A0ABB0 along with its instant centers of rotation P and R

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Fig. 2

Paper and original drawings of the Bresse circles1

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Fig. 1

Jacques Antoine Charles Bresse

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Fig. 6

Fixed and moving axodes for an antiparallelogram mechanism with a = b = 80 deg and c = f = 40 deg: (a) ϕ = 0 deg, (b) ϕ = 30 deg, and (c) ϕ = 90 deg

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Fig. 4

Fixed and moving reference frames

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Fig. 5

Conical fixed axodes (elliptic cones) for a spherical four-bar linkage with a = b = 80 deg and c = f = 40 deg: (a) antiparallelogram mechanism and (b) double-crank mechanism

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Fig. 7

Fixed and moving axodes for an antiparallelogram mechanism with a = b = 60 deg and c = f = 45 deg: (a) ϕ = 0 deg, (b) ϕ = 30 deg, and (c) ϕ = 90 deg

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Fig. 8

Elliptic cone (axode), spherical ellipse (polode), and its evolute curve for a = b = 80 deg and c = f = 70 deg

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Fig. 9

Canonical frame centered at P and composed by the poletangent great circle and polenormal great circle

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Fig. 10

Inflection surface (IS) for Θ=0.4 and inflection spherical cubic for Θ=0.505 : (a) axonometric view of the IS, (b) front-view (xz-plane), (c) side-view (yz-plane), (d) top-view (xy-plane), (e) intersection circles of IS with planes parallel to the xy-plane, and (f) inflection spherical cubic as intersection of the IS with the unit sphere of the spherical motion

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Fig. 11

Inflection spherical cubic and Thales ellipse, along with both polodes and their evolutes: (a) and (b) for a = 67 deg, f = 43 deg, ω = 2 rad/s; (c) and (d) for a = 75 deg, f = 70 deg, ω = 1 rad/s; (e) and (f) for a = 73 deg, f = 71 deg, ω = 1 rad/s



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