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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August 2015
Research-Article
Curvature Theory of Envelope Curve in Two-Dimensional Motion and Envelope Surface in Three-Dimensional Motion
Wei Wang,
Wei Wang
School of Mechanical Engineering,
Linggong Road 2#,
e-mail: wangweidlut@mail.dlut.edu.cn
Dalian University of Technology
,Linggong Road 2#,
Dalian 116024
, China
e-mail: wangweidlut@mail.dlut.edu.cn
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Delun Wang
Delun Wang
1
School of Mechanical Engineering,
Linggong Road 2#,
e-mail: dlunwang@dlut.edu.cn
Dalian University of Technology
,Linggong Road 2#,
Dalian 116024
, China
e-mail: dlunwang@dlut.edu.cn
1Corresponding author.
Search for other works by this author on:
Wei Wang
School of Mechanical Engineering,
Linggong Road 2#,
e-mail: wangweidlut@mail.dlut.edu.cn
Dalian University of Technology
,Linggong Road 2#,
Dalian 116024
, China
e-mail: wangweidlut@mail.dlut.edu.cn
Delun Wang
School of Mechanical Engineering,
Linggong Road 2#,
e-mail: dlunwang@dlut.edu.cn
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. Aug 2015, 7(3): 031019 (9 pages)
Published Online: August 1, 2015
Article history
Received:
December 4, 2012
Revision Received:
November 14, 2014
Online:
May 20, 2015
Citation
Wang, W., and Wang, D. (August 1, 2015). "Curvature Theory of Envelope Curve in Two-Dimensional Motion and Envelope Surface in Three-Dimensional Motion." ASME. J. Mechanisms Robotics. August 2015; 7(3): 031019. https://doi.org/10.1115/1.4029185
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