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Research Papers

Curvature Theory of Envelope Curve in Two-Dimensional Motion and Envelope Surface in Three-Dimensional Motion

[+] Author and Article Information
Wei Wang

School of Mechanical Engineering,
Dalian University of Technology,
Linggong Road 2#,
Dalian 116024, China
e-mail: wangweidlut@mail.dlut.edu.cn

Delun Wang

School of Mechanical Engineering,
Dalian University of Technology,
Linggong Road 2#,
Dalian 116024, China
e-mail: dlunwang@dlut.edu.cn

1Corresponding author.

Manuscript received December 4, 2012; final manuscript received November 14, 2014; published online May 20, 2015. Assoc. Editor: Qiaode Jeffrey Ge.

J. Mechanisms Robotics 7(3), 031019 (Aug 01, 2015) (9 pages) Paper No: JMR-12-1207; doi: 10.1115/1.4029185 History: Received December 04, 2012; Revised November 14, 2014; Online May 20, 2015

The curvature theories for envelope curve of a straight line in planar motion and envelope ruled surface of a plane in spatial motion are systematically presented in differential geometry language. Based on adjoint curve and adjoint surface methods as well as quasi-fixed line and quasi-fixed plane conditions, the centrode and axode are taken as two logical starting-points to study kinematic and geometric properties of the envelope curve of a line in two-dimensional motion and the envelope surface of a plane in three-dimensional motion. The analogical Euler–Savary equation of the line and the analogous infinitesimal Burmester theories of the plane are thoroughly revealed. The contact conditions of the plane-envelope and some common surfaces, such as circular and noncircular cylindrical surface, circular conical surface, and involute helicoid are also examined, and then the positions and dimensions of different osculating ruled surfaces are given. Two numerical examples are presented to demonstrate the curvature theories.

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Figures

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Fig. 1

A line adjoint to the centrodes in planar motion

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Fig. 2

Inflection circle and cuspidal circle for planar motion

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Fig. 3

A plane adjoint to the axodes in spatial motion

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Fig. 4

Position relationship of the generatrix of the envelope surface and ISA

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Fig. 5

Three infinitesimal positions of the plane with β = 0

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Fig. 6

A line L of the coupler link of a crank–rocker linkage

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Fig. 7

Euler–Savary equation for a line of a crank–rocker linkage

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Fig. 8

Spatial RCCC linkage

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