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An Improved Mathematical Model of Galton Board With Velocity-Dependent Restitution

[+] Author and Article Information
Auni Aslah Mat Daud

School of Mathematics and Statistics,
The University of Western Australia,
35 Stirling Highway Crawley,
Perth 6009, Western Australia, Australia
e-mail: auni_aslah@yahoo.com

1Present address: School of Informatics and Applied Mathematics, Universiti Malaysia Terengganu, Kuala Nerus, 21030, Malaysia.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 4, 2016; final manuscript received March 25, 2017; published online September 7, 2017. Assoc. Editor: Tomasz Kapitaniak.

J. Comput. Nonlinear Dynam 12(6), 060901 (Sep 07, 2017) (3 pages) Paper No: CND-16-1178; doi: 10.1115/1.4036418 History: Received April 04, 2016; Revised March 25, 2017

A Galton board is an instrument invented in 1873 by Francis Galton (1822–1911). It is a box with a glass front and many horizontal nails or pins embedded in the back and a funnel. Galton and many modern statisticians claimed that a lead ball descending to the bottom of the Galton board would display random walk. In this study, a new mathematical model of Galton board is developed, to further improve three very recently proposed models. The novel contribution of this paper is the introduction of the velocity-dependent coefficient of restitution. The developed model is then analyzed using symbolic dynamics. The results of the symbolic dynamics analysis prove that the developed Galton board model does not behave the way Galton envisaged.

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References

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Figures

Grahic Jump Location
Fig. 2

A graph of coefficient of restitution as a function of incident speed for colliding objects with same size and different materials, 1 ft = 30.48 cm. Source: Ref. [12].

Grahic Jump Location
Fig. 1

Schematic of a Galton board

Grahic Jump Location
Fig. 3

The fraction of zeros in symbolic sequences versus R. The thick horizontal line at 0.5 indicates the expected fraction of zeros under the binomial random assumption, as assumed by Galton. The thin horizontal lines indicate the 2σ deviations from the expected 0.5 value, and the dashed horizontal lines indicate the 3σ deviations from the expected 0.5 value.

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