Multibody System Dynamics

, Volume 45, Issue 1, pp 105–125 | Cite as

Rocking dynamics of a planar rectangular block on a rigid surface

  • Zhen Zhao
  • Caishan LiuEmail author
  • Nannan Wang


The motion of a rigid rectangular block rocking on a rigid surface involves several motion modes, including rotation around the base corner of the block, point or surface impacts between the block and the surface, and possible uplift following an impact event. With the exception of the surface impact, all other motion states can be easily simulated using the conventional classical mechanics theory. However, the surface impact is closely related to the rapid evolution of the stress field caused by the relative motion of the contact interface, making the associated dynamics relatively complex. Here, we report a novel model of the surface impact that is established by carefully considering the effects from the interface separation and the stick–slip transition of the Coulomb’s friction. This model provides a set of first-order differential equations with respect to the stress impulse, for which numerical methods can be used to obtain the output of the surface impact. In particular, the model can avoid the use of small variables, such as the time of impact and the length of local deformation, thus it is highly beneficial for numerical calculations. We validate this model using the experimental data reported in the existing literature, and then numerically investigate how the material properties of the contact interface affect the rocking motion.


Surface impact Rocking block Friction Darboux–Keller approach Impulsive differential equations 



This work was supported by the NSFC (11572017, 11402033,11702002). We thank Honjian Zhang for providing the detailed experimental data reported in [23, 24] by Dr. Fernando Pena (Instituto de Ingenieria, UNAM Mexico).


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Aeronautic Science and EngineeringBeihang UniversityBeijingChina
  2. 2.State Key Laboratory for Turbulence and Complex Systems, College of EngineeringPeking UniversityBeijingChina
  3. 3.College of EngineeringPeking UniversityBeijingChina

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