A new method is proposed that can deal with multi-impact problems and produce energetically consistent and unique post-impact velocities. A distributing law related to the energy dispersion is discovered by mapping the time scale into the impulsive scale for bodies composed of rate-independent materials. It indicates that the evolution of the kinetic energy during the impacts is closely associated with the relative contact stiffness and the relative potential energy stored at the contact points. This distributing law is combined with the Darboux-Keller method of taking the normal impulse as an independent 'time-like' variable, which obeys a guideline for the selection of an independent normal impulse. Local energy losses are modelled with energetic coefficients of restitution at each contact point. Theoretical developments are presented in the first part in this paper. The second part is dedicated to numerical simulations where numerous and accurate results prove the validity of the approach.
The strong interactions between particles will make the energy within the granular materials propagate through the network of contacts and be partly dissipated. Establishing a model that can clearly classify the dissipation and dispersion effects is crucial for the understanding of the global behaviors in the granular materials. For particles with rate-independent material, the dissipation effects come from the local plastic deformation and can be constrained at the energy level by using energetic restitution coefficients. On the other hand, the dispersion effects should depend on the intrinsic nature of the interaction law between two particles. In terms of a bistiffness compliant contact model that obeys the energetical constraint defined by the energetic coefficients, our recent work related to the issue of multiple impacts indicates that the propagation of energy during collisions can be represented by a distributing law. In particular, this law shows that the dispersion effects are dominated by the relative contact stiffness and the relative potential energy stored at the contact points. In this paper, we will apply our theory to the investigation of the wave behavior in granular chain systems. The comparisons between our numerical results and the experimental ones by Falcon, [Eur. Phys. J. B 5, 111 (1998)] for a column of beads colliding against a wall show very good agreement and confirm some conclusions proposed by Falcon Other numerical results associated with the case of several particles impacting a chain, and the collisions between two so-called solitary waves in a Hertzian type chain are also presented.
Part I of this paper develops a framework that is an extension of the Darboux-Keller shock dynamics towards frictionless multiple impacts between bodies composed of rateindependent materials. A numerical algorithm is proposed in this paper, in which the impulsive differential equation is discretized with respect to the primary impulse corresponding to the contact with the highest potential energy, and the integration step size is estimated at the momentum level. This algorithm respects the energetic constraints and avoids the stiff ordinary differential equation problem arising by directly using the compliant model. The well-known example of Newton's cradle, as well as Bernoulli's system, is used to illustrate the developments.
This paper aims at experimentally investigating the dynamical behaviors when a system of rigid bodies undergoes so-called paradoxical situations. An experimental setup corresponding to the analytical model presented in our prior work [28] is developed, in which a two-link robotic system comes into contact with a moving rail. The experimental results clearly show that a tangential impact does exist at the contact point and takes a peculiar property well coinciding with the maximum dissipation principle stated in [11] by Moreau; the relative tangential velocity of the contact point must immediately approach zero once a Painlevé paradox occurs. After the tangential impact, a bouncing motion may be excited and is influenced by the speed of the moving rail. We adopt the tangential impact rule presented in [28] to determine the post-impact velocities of the system, and use an event-driven algorithm to perform numerical simulations. The qualitative comparisons between the numerical and experimental results are carried out and show good agreements. This study not only presents an experimental support for the shock assumption related to the problem of the Painlevé paradox, but can also find its applications in better understanding the instability phenomena appearing in robotic systems.
The hopping or bouncing motion can be observed when robotic manipulators are sliding on a rough surface. Making clear the reason of generating such phenomenon is important for the control and dynamical analysis for mechanical systems. In particular, such phenomenon may be related to the problem of Painlevé paradox. By using LCP theory, a general criterion for identifying the bouncing motion appearing in a planar multibody system subject to single unilateral constraint is established, and found its application to a two-link robotic manipulator that comes in contact with a rough constantly moving belt. The admissible set in state space that can assure the manipulator keeping contact with the rough surface is investigated, and found which is influenced by the value of the friction coefficient and the configuration of the system. Painlevé paradox can cause either multiple solutions or non-existence of solutions in calculating contact force. Developing some methods to fill in the flaw is also important for perfecting the theory of rigid-body dynamics. The properties of the tangential impact relating to the inconsistent case of Painlevé paradox have been discovered in this paper, and a jump rule for determining the post-states after the tangential impact finishes is developed. Finally, the comprehensively numerical simulation for the two-link robotic manipulator is
The objective of this paper is to implement and test the theory presented in a companion paper for the non-smooth dynamics exhibited in a bouncing dimer. Our approach revolves around the use of rigid body dynamics theory combined with constraint equations from the Coulomb's frictional law and the complementarity condition to identify the contact status of each contacting point. A set of impulsive differential equations based on Darboux-Keller shock dynamics is established that can deal with the complex behaviours involved in multiple collisions, such as the frictional effects, the local dissipation of energy at each contact point, and the dispersion of energy among various contact points. The paper will revisit the experimental phenomena found in Dorbolo et al. (Dorbolo et al. 2005 Phys. Rev. Lett. 95, 044101), and then present a qualitative analysis based on the theory proposed in part I. The value of the static coefficient of friction between the plate and the dimer is successfully estimated, and found to be responsible for the formation of the drift motion of the bouncing dimer. Plenty of numerical simulations are carried out, and precise agreements are obtained by the comparisons with the experimental results.
This paper concerns a disc-ball system, in which a moving ball collides against a disc resting on a rough, fixed horizontal surface. The complexity in such a simple object is due to the presence of the line contact between the disc and the fixed plate, which significantly influences the impact-generated state of the disc. We deal with this problem in a uniform framework that encapsulates different structures of the mathematical model, including contacts, impacts, stick-slip in friction, as well as the transitions among different states of the variablestructure dynamics. We design specific experiments that provide useful information to help determine the macroscopic parameters in impact and friction. Other complicated cases concerned with the couplings between impacts and friction are theoretically and experimentally investigated. Excellent agreements between numerical and experimental results support our theoretical developments.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.