Variable structure dynamics in a bouncing dimer

Systems with unilateral constraints usually possess a variable structure dynamics, in which systems can switch from one mode to another due to contacts and impacts. In particular, friction participating into contacts and impacts will extremely complicate the dynamics, and even result in some singularities when using rigid body models. In this paper, we develop a method that can well deal with the difficulties involved in the variable structure systems, such as the multiple impacts with friction (i.e. the occurrence of simultaneous impacts), the superstatic problems in rigid body systems, the multivalued graph in the stick mode of the Coulomb’s friction, and the inelastic collapse in impacts. The transition rules to monitor the switches from one mode to another are also established, in which the normal states at each contact point are controlled by the complementary conditions for a contact process and by the potential energy at contact points for an impact process, while the tangential states at the contact point will be governed by a correlative coefficient of friction defined by the tangential differential equations. A system elaborated in [38] with precise experimental results serves as an example to illustrate the theoretical developments, in which a dimer consisting of two spheres rigidly connected by a light glass rod bounces on a vibrating plate. This system, even though simple enough, exhibits profuse physical phenomena under different initial and driving conditions, and may spur different ordered persistent motions, such as the drift, jump and flutter modes. In particular, each mode of the persistent motion is synthesized by a periodically complicated motion that may involve single and double impacts, contacts with or without slip, etc. Based on the theory proposed in this paper, we clearly explain the regime of persistent motions in the dimer and find that the peculiar property of friction with discontinuity plays a significant role for its formation. Plenty of numerical simulations are carried out, and precise agreements between the numerical and experimental results are obtained. Furthermore, a simplified model for the dimer in drift mode is developed and theoretical analysis is implemented. An approximate formula for the mean horizonal velocity is obtained that also coincides well with the experimental findings. This may be beneficial for the study ∗ State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing, China, 100871 † INRIA, Bipop Research Team, ZIRST Montbonnot, 655 Avenue de l’Europe, F-38334 Saint-Ismier, France in ria -0 03 37 48 2, v er si on 1 7 N ov 2 00 8 2 Liu, Zhao & Brogliato of complex systems dynamics, in which there exists an intrinsic connection between the ordered behaviors of the systems and the microsize parameters of its ingredients. Key-words: nonsmooth mechanical system, impact, Coulomb friction, complex systems, numerical simulation. INRIA in ria -0 03 37 48 2, v er si on 1 7 N ov 2 00 8 Dynamique à structure variable d’un dimère rebondissant Résumé : Les systèmes avec contraintes uniltérales possèdent une structure variable, avec plusieurs modes définis par les onctacts avec ou sans frottement et les impacts. Dans cet article nous proposons une mt́hode num‘erique associée à un modèle d’impacts multiples avec frottement, afin de simuler un dimère qui rebondit sur une table vibrante. Un tel système, bien que relativement simple, incorpore la plupart des difficultés liées au caractère non-régulier des systèmes avec contact unilatéral frotant. Les résultats numériques sont comparés avec succès aux résultats expérimentaux présentés par Dorbolo et al. (Physical Review Letters, 95, 2005). Mots-clés : mécanique non-régulière, impacts, frottement de Coulomb, simulation numérique, systèmes complexes. in ria -0 03 37 48 2, v er si on 1 7 N ov 2 00 8 4 Liu, Zhao & Brogliato