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Research Papers

Toppling Dynamics of Regularly Spaced Dominoes in an Array

[+] Author and Article Information
Tengfei Shi

State Key Laboratory for Turbulence
and Complex Systems,
College of Engineering Peking University,
Beijing 100871, China

Yang Liu

College of Engineering,
Mathematics and Physical
Sciences University of Exeter,
North Park Road,
Exeter EX4 4QF, UK

Nannan Wang

State Key Laboratory for Turbulence and
Complex Systems,
College of Engineering Peking University,
Beijing 100871, China

Caishan Liu

State Key Laboratory for Turbulence and
Complex Systems,
College of Engineering Peking University,
Beijing 100871, China
e-mail: liucs@pku.edu.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received November 22, 2017; final manuscript received January 15, 2018; published online February 9, 2018. Assoc. Editor: Ahmet S. Yigit.

J. Appl. Mech 85(4), 041008 (Feb 09, 2018) (10 pages) Paper No: JAM-17-1648; doi: 10.1115/1.4039047 History: Received November 22, 2017; Revised January 15, 2018

This paper studies a new comprehensive model for toppling dynamics of regularly spaced dominoes in an array. The model has unlocked the hypotheses introduced by Stronge and Shu (Stronge, W. J., and Shu, D., 1988, “The Domino Effect: Successive Destabilization by Cooperative Neighbours,” Proc. R. Soc. A, 418(1854), pp. 155–163), which can provide us some essential insights into the mechanism of domino wave. Extensive comparisons are made between the proposed model and the experimental results studied in the existing literature. Our numerical studies show that the existing theoretical models are special cases of the proposed model, and the fluctuation in the waveform of propagation speed observed from experiments was caused by the irregular multiple impacts between colliding dominoes. The influence of physical parameters of domino on the natural speed of toppling dominoes is also considered, and it is found that the coefficients of friction and restitution between colliding dominoes have more effects due to the energy dissipation during impact.

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References

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Figures

Grahic Jump Location
Fig. 1

Toppling dominoes on the ground

Grahic Jump Location
Fig. 2

Multipoints contact between the κth block and the ground

Grahic Jump Location
Fig. 3

Dimensionless propagation speed v̂ varies as a function of the number of dominoes n for different initial toppling speeds. Parameters of Domino X in Table 1 were used. To compare with experimental results, our calculations were carried out using a fast initial dimensionless angular speed at 1.8 and a slow one at 0.5. Simulation results for fast and slow initial toppling angular speeds are marked by and , and the experimental results obtained from Ref. [17] for fast and slow initial toppling angular speeds are marked by • and , respectively.

Grahic Jump Location
Fig. 4

Dimensionless speed v̂na varies as a function of the domino spacing s/h for Dominoes T and P in Table 1. Simulation results using the proposed domino model are marked by and , and the experimental results obtained from Ref. [13] are given by and . Theoretical solution for Dominoes T and P calculated using Stronge's cooperative group model is denoted by the solid and dot-dash line, respectively.

Grahic Jump Location
Fig. 5

Dimensionless natural speed v̂na varies as a function of the domino spacing s/h for Domino T in Table 1. In all of the simulations, the material parameters between domino and ground are always set to e2=0 and μ2=0.9. Numerical results obtained by using different material parameters between dominos are plotted as follows: ■ (e1=0,μ1=0); and  (μ1=0.15,e1=0). Theoretical solutions calculated by the cooperative group model, the single collision model, and the extended group model are denoted by solid line, dot-dash line, and dash line, respectively.

Grahic Jump Location
Fig. 6

Snapshots of the toppling of a domino array separated by a spacing of s/h=0.52. Simulations were performed by using the parameters of Domino X given in Table 1 for (a) μ1=0.17, e1=0.85, μ2=0.25 and e2=0.5; (b) μ1=0, e1=0, μ2=0.9 and e2=0.

Grahic Jump Location
Fig. 7

Dimensionless angular velocity θ˙̂i of 7th, 8th, 9th, and 10th block as a function of dimensionless time t/t¯ obtained from simulations by using the parameters of Domino X given in Table 1 for (a) μ1=0.17, e1=0.85, μ2=0.25 and e2=0.5; (b) μ1=0, e1=0, μ2=0.9 and e2=0. Time t = 0 is defined as the moment when the domino forms a contact with its left neighbor.

Grahic Jump Location
Fig. 8

Dimensionless propagation speed v̂ varies as a function of the number of dominoes n calculated for parameters of Domino X in Table 1 for (a) μ1=0.17, e1=0.85, μ2=0.25 and e2=0.5; (b) μ1=0, e1=0, μ2=0.9 and e2=0 under different initial toppling speeds cases

Grahic Jump Location
Fig. 9

Dimensionless natural speed v̂na as a function of (a) coefficient of friction μ1 calculated for d/h=0.18, μ2=0.9, e1=0 and e2=0; (b) coefficient of restitution e1 calculated for d/h=0.18, μ2=0.9, e2=0 and μ1=0, for different relative spacing s/d

Grahic Jump Location
Fig. 10

Dimensionless natural speed v̂na as a function of (a) coefficient of friction μ2 calculated for d/h=0.18, μ1=0, e1=0 and e2=0; (b) coefficient of restitution e2 between domino and ground calculated for d/h=0.18, e1=0, μ2=0.9 and μ1=0, for different relative spacing s/d

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