0
Technical Brief

Design of Mechanisms to Draw Trigonometric Plane Curves

[+] Author and Article Information
Yang Liu

Robotics and Automation Laboratory,
Department of Mechanical and Aerospace Engineering,
University of California,
Irvine, CA 92697
e-mail: liuy14@uci.edu

J. Michael McCarthy

Professor
Fellow ASME
Robotics and Automation Laboratory,
Department of Mechanical and Aerospace Engineering,
University of California,
Irvine, CA 92697
e-mail: jmmccart@uci.edu

Manuscript received October 20, 2016; final manuscript received January 13, 2017; published online March 9, 2017. Assoc. Editor: Venkat Krovi.

J. Mechanisms Robotics 9(2), 024503 (Mar 09, 2017) (8 pages) Paper No: JMR-16-1328; doi: 10.1115/1.4035882 History: Received October 20, 2016; Revised January 13, 2017

This paper describes a mechanism design methodology that draws plane curves which have finite Fourier series parameterizations, known as trigonometric curves. We present three ways to use the coefficients of this parameterization to construct a mechanical system that draws the curve. One uses Scotch yoke mechanisms for each of the terms in the coordinate trigonometric functions, which are then added using a belt or cable drive. The second approach uses two-coupled serial chains obtained from the coordinate trigonometric functions. The third approach combines the coordinate trigonometric functions to define a single-coupled serial chain that draws the plane curve. This work is a version of Kempe's universality theorem that demonstrates that every plane trigonometric curve has a linkage which draws the curve. Several examples illustrate the method including the use of boundary points and the discrete Fourier transform to define the trigonometric curve.

Copyright © 2017 by ASME
Topics: Chain
Your Session has timed out. Please sign back in to continue.

References

Coros, S. , Thomaszewski, B. , Noris, G. , Sueda, S. , Forberg, M. , Sumner, R. W. , Matusik, W. , and Bickel, B. , 2013, “ Computational Design of Mechanical Characters,” ACM Trans. Graphics, 32(4), p. 83. [CrossRef]
Thomaszewski, B. , Coros, S. , Gauge, D. , Megaro, V. , Grinspun, E. , and Gross, M. , 2014, “ Computational Design of Linkage-Based Characters,” ACM Trans. Graphics, 33(4), p. 64. [CrossRef]
Nolle, H. , 1974, “ Linkage Coupler Curve Synthesis: A Historical Review—II: Developments After 1875,” Mech. Mach. Theory, 9(3–4), pp. 325–348. [CrossRef]
Nolle, H. , 1974, “ Linkage Coupler Curve Synthesis: A Historical Review—I: Developments Up To 1875,” Mech. Mach. Theory, 9(2), pp. 147–168. [CrossRef]
Koetsier, T. , 1983, “ A Contribution to the History of Kinematics—I: Watt's Straight-Line Linkages and the Early French Contributions to the Theory of the Planar 4-Bar Coupler Curve,” Mech. Mach. Theory, 18(1), pp. 37–42. [CrossRef]
Koetsier, T. , 1983, “ A Contribution to the History of Kinematics—II: The Work of English Mathematicians on Linkages During the Period 1869-78,” Mech. Mach. Theory, 18(1), pp. 43–48. [CrossRef]
Kempe, A. B. , 1876, “ On a General Method of Describing Plane Curves of the nth Degree by Linkwork,” Proc. London Math. Soc., VII(102), pp. 213–216.
Kempe, A. B. , 1877, How to Draw a Straight Line, Macmillan, London.
Jordan, D. , and Steiner, M. , 1999, “ Configuration Spaces of Mechanical Linkages,” Discrete Comput. Geom., 22(2), pp. 297–315. [CrossRef]
Kapovich, M. , and Millson, J. J. , 2002, “ Universality Theorems for Configuration Spaces of Planar Linkages,” Topology, 41(6), pp. 1051–1107. [CrossRef]
Kobel, A. , 2008, “ Automated Generation of Kempe Linkages for Algebraic Curves in a Dynamic Geometry System,” BCS thesis, Saarland University, Saarbrucken, Germany.
Saxena, A. , 2011, “ Kempe's Linkages and the Universality Theorem,” Resonance, 16(220), pp. 220–237. [CrossRef]
Liu, Y. , and McCarthy, J. M. , 2017, “ Synthesis of a Linkage to Draw a Plane Algebraic Curve,” Mech. Mach. Theory, 111, pp. 10–20. [CrossRef]
Hong, H. , and Schicho, J. , 1998, “ Algorithms for Trigonometric Curves (Simplification, Implicitization, Parameterization),” J. Symbolic Comput., 26(3), pp. 279–300. [CrossRef]
Artobolevskii, I. I. , 1964, Mechanisms for the Generation of Plane Curves, Pergamon Press, London.
Miller, D. C. , 1916, “ A 32-Element Harmonic Synthesizer,” J. Franklin Inst., 181(1), pp. 51–81. [CrossRef]
Nie, X. , and Krovi, V. , 2005, “ Fourier Methods for Kinematic Synthesis of Coupled Serial Chain Mechanisms,” ASME J. Mech. Des., 127(2), pp. 232–241. [CrossRef]
Oppenheim, A. V. , and Schafer, R. W. , 2010, Discrete-Time Signal Processing, Pearson Higher Education, Upper Saddle River, NJ.
Kendig, K. , 2011, A Guide to Plane Algebraic Curves, MAA, Washington, DC.
Shikin, E. V. , 1995, Handbook and Atlas of Curves, CRC Press, Boca Raton, FL.
Fay, T. H. , 1989, “ The Butterfly Curve,” Am. Math. Mon., 96(5), pp. 442–443. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Artobolevskii [15] designed this mechanism to draw the trifolium (three-petal) curve

Grahic Jump Location
Fig. 2

A Scotch yoke mechanism that transforms the rotation of a crank into a cosine curve

Grahic Jump Location
Fig. 3

One set of Scotch yoke mechanisms drives the x-component and another set drives the y-component of a cursor to draw a plane curve

Grahic Jump Location
Fig. 4

A coupled serial chain drives the x-component and a separate coupled serial chain drives the y-component of a cursor to draw a plane curve

Grahic Jump Location
Fig. 5

The end-point of a single-coupled serial chain draws a plane curve

Grahic Jump Location
Fig. 6

Trifolium with the size of the petal set to a = 2

Grahic Jump Location
Fig. 7

A system of Scotch yoke mechanisms driven by a single input that draws the trifolium

Grahic Jump Location
Fig. 8

A system of two-coupled serial chains driven by a single input that draws the trifolium

Grahic Jump Location
Fig. 9

A single constrained coupled serial chain with one input that draws the trifolium

Grahic Jump Location
Fig. 10

The plot of a butterfly curve from its polar equation

Grahic Jump Location
Fig. 11

The constraint coupled serial chain to draw this butterfly curve consists of 14 terms

Grahic Jump Location
Fig. 12

The shape of the Batman logo

Grahic Jump Location
Fig. 13

Batman logo curve obtained using 20 terms of the discrete Fourier transform of boundary points

Grahic Jump Location
Fig. 14

The system of component Scotch yoke mechanisms that draw the Batman logo consists of two sets of 19 mechanisms

Grahic Jump Location
Fig. 15

The single-coupled serial chain that draws the Batman logo consists of 38 links

Grahic Jump Location
Fig. 16

A photograph of the component parts of the trifolium drawing mechanism that were manufactured from acrylonitrile butadiene styrene using additive manufacture

Grahic Jump Location
Fig. 17

A photograph of the assembled mechanism that draws the trifolium

Grahic Jump Location
Fig. 18

The photograph shows the trifolium traced by the trajectory of a LED positioned at the drawing point of the mechanism

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In