# Toppling dynamics of a mass-varying domino system

Original paper

## Abstract

This paper studies the toppling dynamics of a mass-varying domino system for which the mass of the domino changes at an exponential rate of its sequence number. By introducing geometrical constraints representing the interactions between the dominoes and the ground, we propose a simplified model which can describe their toppling dynamics as a one-degree-of-freedom structural-varying system. Principles of energy and generalized momentum conservation are used to investigate the free falling and the colliding phases of the system, and our proposed model is validated through a comprehensive numerical study. Furthermore, we develop an impact mapping for studying the evolution of the system by using the mathematical properties of their geometrical constraints and establish the occurrence conditions for solitary wave. According to these conditions, the system can exhibit four different propagation modes: non-solitary mode, uniform solitary mode, accelerating solitary mode and stopped mode. Based on the studies in this paper, we can reveal the dynamic characteristics of domino phenomenon and provide an insight into its energy evolution.

## Keywords

Domino system Domino phenomenon Rigid body dynamics Wave propagation

## List of symbols

d

Thickness of domino (m)

e

Coefficient of restitution (−)

E

Mechanical energy (J)

$$F^n$$

Normal contact force (N)

$$F\tau$$

Tangential contact force (N)

$$\mathcal {F}_I$$

Inertial coordinate frame (−)

$$\mathcal {F}_i$$

Body fixed coordinate frame (−)

g

Gravitational acceleration (m s$$^{-2}$$)

h

Height of domino (m)

$$\mathbf {i, j},\mathbf {i}_i, \mathbf {j}_i$$

Unit vectors in coordinate systems (−)

i, j, n

Sequence index (−)

$$I^c$$

Moment inertia about mass centre (kg m$$^{2}$$)

I

Moment inertia about lower right corner (kg m$$^{2}$$)

$$\lambda$$

Number of the discrete points on contact interface (−)

k

Attenuation exponent of domino’s motion (−)

$$m_i$$

Mass of ith domino (kg)

q

Mass exponent (−)

$$\mathbf {q}$$

Column vector of generalized coordinates (−)

s

Spacing between dominoes (m)

t

Time (s)

T

Kinetic energy (J)

u

Dimensionless height of the centre of mass (−)

$$v^\tau$$

Relative tangential velocity (m s$$^{-1}$$)

V

Gravitational potential energy (J)

x, y

Position of domino’s centre (m)

$$\theta _i$$

Rotation angle of ith domino (rad)

$$\theta _c$$

$$\theta _m$$

$$\delta$$

Relative normal displacement (m)

$$\mu$$

Slip friction coefficient (−)

$$\mu ^s$$

Stick friction coefficient (−)

$$\varPsi ^i_n$$

Angle of ith domino when the nth domino’s angle is $$\theta$$ (rad)

$$\varPhi ^i_n$$

Ratio of the angular velocities between the ith and the nth dominoes (−)

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## Authors and Affiliations

• Tengfei Shi
• 1
• Yang Liu
• 2
• Nannan Wang
• 1
• Caishan Liu
• 1
1. 1.State Key Laboratory for Turbulence and Complex Systems, College of EngineeringPeking UniversityBeijingChina
2. 2.College of Engineering, Mathematics and Physical SciencesUniversity of ExeterExeterUK