Toppling dynamics of a mass-varying domino system

  • Tengfei Shi
  • Yang Liu
  • Nannan Wang
  • Caishan LiuEmail author
Original paper


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.


Domino system Domino phenomenon Rigid body dynamics Wave propagation 

List of symbols


Thickness of domino (m)


Coefficient of restitution (−)


Mechanical energy (J)


Normal contact force (N)

\(F\tau \)

Tangential contact force (N)

\(\mathcal {F}_I\)

Inertial coordinate frame (−)

\(\mathcal {F}_i\)

Body fixed coordinate frame (−)


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


Height of domino (m)

\(\mathbf {i, j},\mathbf {i}_i, \mathbf {j}_i\)

Unit vectors in coordinate systems (−)

i, j, n

Sequence index (−)


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


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

\(\lambda \)

Number of the discrete points on contact interface (−)


Attenuation exponent of domino’s motion (−)


Mass of ith domino (kg)


Mass exponent (−)

\(\mathbf {q}\)

Column vector of generalized coordinates (−)


Spacing between dominoes (m)


Time (s)


Kinetic energy (J)


Dimensionless height of the centre of mass (−)

\(v^\tau \)

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


Gravitational potential energy (J)

x, y

Position of domino’s centre (m)

\(\theta _i\)

Rotation angle of ith domino (rad)

\(\theta _c\)

Domino’s spacing angle (rad)

\(\theta _m\)

Domino’s maximum tilt angle (rad)

\(\delta \)

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 (−)



This work has been supported by the National Natural Science Foundation of China under Grant Nos. 11702002, 11932001. The authors would like to specially thank Professor Jianhua Xie for providing inspirations.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest concerning the publication of this manuscript.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  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

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