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This research is focused on improving the solutions obtained using theory in contact and impact modeling. A theoretical framework is developed which can simulate the performance of dynamic systems within a real world environment. This environment involves conditions, such as contact, impact and friction. Numerical simulation provides an easy way to perform numerous iterations with varying conditions, which is more cost effective than building equivalent experimental setups. The developed framework will serve as a tool for engineers and scientists to gain some insight on predicting how a system may behave. The current field of research in multibody system dynamics lacks a framework for modeling simultaneous, indeterminate contact and impact with friction. This special class of contact and impact problems is the major focus of this research. This research develops a framework, which contributes to the existing literature. The contact and impact problems examined in this work are indeterminate with respect to the impact forces. This is problematic because the impact forces are needed to determine the slip-state of contact and impact points. The novelty of the developed approach relies on the formation of constraints among the velocities of the impact points. These constraints are used to address the indeterminate nature of the collisions encountered. This approach strictly adheres to the assumptions of rigid body modeling in conjunction with the notion that the configuration of the system does not change in the short time span of the collision. These assumptions imply that the impact Jacobian is constant during the collision, which enforces a kinematic relationship between the impact points. The developed framework is used to address simultaneous, indeterminate contact and impact problems with friction. In the preliminary stages of this research, an iterative method, which incorporated an optimization function was used obtain the solutions for numerical solution to the collision. In an effort to improve the time and accuracy of the results, the iterative method was replaced with an analytical approach and implemented with the constraint formulation to achieve more energetically consistent solutions (i.e. there are no unusual gains in energy after the impact). The details of why this claim is valid will be discussed in more detail in this dissertation. The analytical framework was developed for planar contact and impact problems, while a numerical framework is developed for three-dimensional (3D) problems. The modeling of friction in 3D presents some challenging issues that are well documented in the literature, which make it difficult to apply an analytical framework. Simulations are conducted for a planar ball, planar rocking block problem, Newton's Cradle, 3D sphere, and 3D rocking block. Some examples serve as benchmark problems, in which the results are validated using experimental data.
Contact and impact analyses are an essential part of multibody dynamic simulations.Modeling of contact and impact problems have applications in a wide variety of areas including robotics, earthquake engineering, computer graphics, and manufacturing. Collisions between objects typically take place over surfaces that are represented by a set of points in the operation space, thereby requiring multi-point contact and impact analysis.Analysis of multi-point contact and impact may lead to indeterminate (underdetermined) problems with more number of unknowns (contact forces) than equations.This work pertains to the problem of resolving multi-point contact and impact problems in multibody systems consisting of hard objects, that can be assumed to be rigid.In the first part of this work, a rigidity based modeling and simulation technique is developed for multi-point impacts between hard objects. In this proposed framework impacts are treated as discrete events during which the velocities of the system evolve in the impulse-domain, based on an impulse-momentum theory called Darboux-Keller shock.Constraints derived based on the rigid body assumption are used to resolve indeterminacy associated with multi-point analysis. An energetic terminal constraint is also proposed. based on Stronge's Hypothesis, that guarantees the treatment of impact to be energetically consistent. This approach is used to derive both planar and three-dimensional models of multi-point indeterminate impacts.The rigid impact model based on impulse-momentum theory, developed in the first part of this work, loses some information like force and deformation histories during impacts. This lost information, however can be useful in certain types of application. Hence, to retain this information, the second part of this work proposes a method of augmenting the rigid-impact model with a contact force model from the contact mechanics literature to simultaneously determine the force and deformation histories during an impact event.The contact force model used here is a viscoelastoplastic model of contact that considers the effects of permanent (plastic) deformation in the material. A relationship is developed between the permanent deformations of the material and the energetic terminal constraint proposed in the first part of this work to characterize the force histories during collisions.The accumulation of discrete impact events during the time-domain simulation may lead to chattering or zeno phenomenon, causing the adaptive step-size integration to halt or fail. This work resolves this problem by transitioning to contact when the normal components of the post-impact velocities become very small. During contact, the forces between the participating rigid bodies satisfy the: 1) non-penetrability condition and 2) frictional force constraints based on Coulomb Friction. The non-penetrability condition enforces normal velocity and acceleration constraints on the equations of motion, whereas the Coulomb friction constrains the tangential forces at the contact points. These constraints placed on the equations of motion, lead to a reduction in the number of degrees of freedom (DOF) of the system. This work uses an online constraint embedding technique to enforce contact constraints.
This work presents a method for understanding the impact behaviour of a flexible body undergoing multiple, simultaneous contacts. A continuous model is used with an event-driven function in MATLAB, which detects the collisions. The flexible body is defined as a system of particles, having inter-particle forces in terms of spring, damper coefficients. Equations of motion for such a Flexible MultiBody system are determined and then solved for different phases. In the method presented, the indeterminate nature of equations of motion encountered, during impact, and contact for flexible body are examined. Constraint forces are determined during the different phases of an impact to address the equations. These techniques are applied to a planar model of an elliptical body, which is dropped freely under the effect of gravity and collision occurs at the ground determined. A simulation is presented demonstrating the behaviour of the body during impact, and contact with the ground with the proposed method.
The volume introduces basic concepts necessary for a modern treatment of inequality problems in finite degree of freedom dynamics. Tools from convex analysis, by now well established in non-smooth mechanics, are used to formulate the constitutive equations and impact laws. The lectures cover a broad area of non-smooth dynamics from primal and dual energy functions in variational and differential form to application problems as chimney dampers or vibration conveyors. This includes frictional oscillations with bifurcation scenarios as well as analogies to small displacement quasi-static problems. The course is on an advanced level, designed primarily for postgraduate students, but should also be of value for scientists working on dynamic complementarity problems.
This work presents a method for determining the post-impact behavior of a rigid-body undergoing simultaneous, multiple impacts with friction. A discrete algebraic model is used with an event-driven function which finds impact events. In this work, the indeterminate nature of the equations of motion encountered at impact are examined. A velocity constraint is developed based on the rigid-body assumption to address the equations and an impact law is used to determine the impulsive forces. The slip-state of each contact point is then determined and appropriate methods are used to resolve the post-impact velocities. Friction is treated as a complementarity problem and a set complementarity conditions are formulated using Coulomb's friction law. Additional constraints are composed in terms of a dissipation principle to yield a solution for the post-impact tangential velocities. These works will be applied to a simple planar model of a ball which is forced to impact a corner between the ground and a wall. Computer simulations will be presented to demonstrate the post- impact behavior of a rigid-body which experiences simultaneous, multiple impacts with friction.
This volume provides the international multibody dynamics community with an up-to-date view on the state of the art in this rapidly growing field of research which now plays a central role in the modeling, analysis, simulation and optimization of mechanical systems in a variety of fields and for a wide range of industrial applications. This book contains selected contributions delivered at the ECCOMAS Thematic Conference on Multibody Dynamics, which was held in Brussels, Belgium and organized by the Université catholique de Louvain, from 4th to 7th July 2011. Each paper reflects the State-of-Art in the application of Multibody Dynamics to different areas of engineering. They are enlarged and revised versions of the communications, which were enhanced in terms of self-containment and tutorial quality by the authors. The result is a comprehensive text that constitutes a valuable reference for researchers and design engineers which helps to appraise the potential for the application of multibody dynamics methodologies to a wide range of areas of scientific and engineering relevance.
This book analyzes several compliant contact force models within the context of multibody dynamics, while also revisiting the main issues associated with fundamental contact mechanics. In particular, it presents various contact force models, from linear to nonlinear, from purely elastic to dissipative, and describes their parameters. Addressing the different numerical methods and algorithms for contact problems in multibody systems, the book describes the gross motion of multibody systems by using a two-dimensional formulation based on the absolute coordinates and employs different contact models to represent contact-impact events. Results for selected planar multibody mechanical systems are presented and utilized to discuss the main assumptions and procedures adopted throughout this work. The material provided here indicates that the prediction of the dynamic behavior of mechanical systems involving contact-impact strongly depends on the choice of contact force model. In short, the book provides a comprehensive resource for the multibody dynamics community and beyond on modeling contact forces and the dynamics of mechanical systems undergoing contact-impact events.
This volume contains the edited version of selected papers presented at the Nato Advanced Study Institute on "Computer Aided Analysis of Rigid and Flexible Mechanical Systems", held in Portugal, from the 27 June to 9 July, 1994. The present volume can be viewed as a natural extension of the material addressed in the Institute which was published by KLUWER in the NATO ASI Series, Vol. 268, in 1994. The requirements for accurate and efficient analysis tools for design of large and lightweight mechanical systems has driven a strong interest in the challenging problem of multibody dynamics. The development of new analysis and design formulations for multi body systems has been more recently motivated with the need to include general features such as: real-time simulation capabilities, active control of machine flexibilities and advanced numerical methods related to time integration of the dynamic systems equations. In addition to the presentation of some basic formulations and methodologies in dynamics of multibody systems, including computational aspects, major applications of developments to date are presented herein. The scope of applications is extended to vehicle dynamics, aerospace technology, robotics, mechanisms design, intermittent motion and crashworthiness analysis. Several of these applications are explored by many contributors with a constant objective to pace development and improve the dynamic performance of mechanical systems avoiding different mechanical limitations and difficult functional requirements, such as, for example, accurate positioning of manipulators.
This book has evolved from the passionate desire of the authors in using the modern concepts of multibody dynamics for the design improvement of the machineries used in the rural sectors of India and The World. In this connection, the first author took up his doctoral research in 2003 whose findings have resulted in this book. It is expected that such developments will lead to a new research direction MuDRA, an acronym given by the authors to “Multibody Dynamics for Rural Applications. ” The way Mu- DRA is pronounced it means ‘money’ in many Indian languages. It is hoped that practicing MuDRA will save or generate money for the rural people either by saving energy consumption of their machines or making their products cheaper to manufacture, hence, generating more money for their livelihood. In this book, the initial focus was to improve the dynamic behavior of carpet scrapping machines used to wash newly woven hand-knotted c- pets of India. However, the concepts and methodologies presented in the book are equally applicable to non-rural machineries, be they robots or - tomobiles or something else. The dynamic modeling used in this book to compute the inertia-induced and constraint forces for the carpet scrapping machine is based on the concept of the decoupled natural orthogonal c- plement (DeNOC) matrices. The concept is originally proposed by the second author for the dynamics modeling and simulation of serial and - rallel-type multibody systems, e. g.
The extension of collision models for single impacts between two bodies, to the case of multiple impacts (which take place when several collisions occur at the same time in a multibody system) is a challenge in Solid Mechanics, due to the complexity of such phenomena, even in the frictionless case. This monograph aims at presenting the main multiple collision rules proposed in the literature. Such collisions typically occur in granular materials, the simplest of which are made of chains of aligned balls. These chains are used throughout the book to analyze various multiple impact rules which extend the classical Newton (kinematic restitution), Poisson (kinetic restitution) and Darboux-Keller (energetic or kinetic restitution) approaches for impact modelling. The shock dynamics in various types of chains of aligned balls (monodisperse, tapered, decorated, stepped chains) is carefully studied and shown to depend on several parameters: restitution coefficients, contact stiffness ratios, elasticity coefficients (linear or nonlinear force/ indentation relation), and kinetic angles (that depend on the mass ratios). The dissipation and the dispersion of kinetic energy during a multiple impact are mandatory modelling, and are quantified with suitable indices. Particular attention is paid to the ability of the presented laws to correctly predict the wave effects in the chains. Comparisons between many numerical and experimental results are shown, as well as comparisons between four different impact laws in terms of their respective abilities to correctly model dissipation and dispersion of energy.