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Multibody Dynamics Computational Methods in Applied Sciences Volume 12 Series Editor E Oñate International Center for Numerical Methods in Engineering (CIMNE) Technical University of Catalunya (UPC) Edif cio C-1, Campus Norte UPC Gran Capitán, s/n 08034 Barcelona, Spain onate@cimne.upc.edu www.cimne.com For other titles published in this series, go to www.springer.com/series/6899 Carlo L Bottasso Multibody Dynamics Computational Methods and Applications 123 Editor C.L Bottasso Politecnico di Milano Dipartimento di Ingegneria Aerospaziale Via La Masa, 34 20156 Milano Italy carlo.bottasso@polimi.it ISBN 978-1-4020-8828-5 e-ISBN 978-1-4020-8829-2 Library of Congress Control Number: 2008933572 All Rights Reserved c 2009 Springer Science + Business Media B.V No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Printed on acid-free paper springer.com Preface Multibody Dynamics is an area of Computational Mechanics which blends together various disciplines such as structural dynamics, multi-physics mechanics, computational mathematics, control theory and computer science, in order to deliver methods and tools for the virtual prototyping of complex mechanical systems Multibody dynamics plays today 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 The ECCOMAS Thematic Conference on Multibody Dynamics was initiated in Lisbon in 2003, and then continued in Madrid in 2005 with the goal of providing researchers in Multibody Dynamics with appropriate venues for exchanging ideas and results The third edition of the Conference was held at the Politecnico di Milano, Milano, Italy, from June 25 to June 28, 2007 The Conference saw the participation of over 250 researchers from 32 different countries, presenting 209 technical papers, and proved to be an excellent forum for discussion and technical exchange on the most recent advances in this rapidly growing field This book is a collection of revised and expanded versions of papers presented at the Conference Goal of this collection of works is to offer an upto-date view on some of the most recent cutting edge research developments in Multibody Dynamics Contributions have been selected from all sessions of the Conference, and cover the areas of biomechanics (Ackermann and Schiehlen, Millard et al.), contact dynamics (Tasora and Anitescu), control, mechatronics and robotics (Bottasso), flexible multibody dynamics (Cugnon et al., Lunk and Simeon, Betsch and Să anger), formulations and numerical methods (Jay and Negrut), optimization (Collard et al.), real-time simulation (Binami et al.), software development, validation, education (Pennestr`ı and Valentini), and vehicle systems (Ambr´ osio et al.) I hope you will find the reading of this collection enjoyable and stimulating, as we anxiously wait for the 2009 edition of this excellent Conference Series Milano, May 2008 Carlo L Bottasso V Contents Preface V Physiological Methods to Solve the Force-Sharing Problem in Biomechanics Marko Ackermann and Werner Schiehlen Multi-Step Forward Dynamic Gait Simulation Matthew Millard, John McPhee, and Eric Kubica 25 A Fast NCP Solver for Large Rigid-Body Problems with Contacts, Friction, and Joints Alessandro Tasora and Mihai Anitescu 45 Solution Procedures for Maneuvering Multibody Dynamics Problems for Vehicle Models of Varying Complexity Carlo L Bottasso 57 Synthesis and Optimization of Flexible Mechanisms Frederic Cugnon, Alberto Cardona, Anna Selvi, Christian Paleczny, and Martin Pucheta 81 The Reverse Method of Lines in Flexible Multibody Dynamics Christoph Lunk and Bernd Simeon 95 A Nonlinear Finite Element Framework for Flexible Multibody Dynamics: Rotationless Formulation and Energy-Momentum Conserving Discretization Peter Betsch and Nicolas Să anger 119 A Second Order Extension of the Generalized–α Method for Constrained Systems in Mechanics Laurent O Jay and Dan Negrut 143 VII VIII Contents Kinematical Optimization of Closed-Loop Multibody Systems Jean-Fran¸cois Collard, Pierre Duysinx, and Paul Fisette 159 A Comparison of Three Different Linear Order Multibody Dynamics Algorithms in Limited Parallel Computing Environments Adarsh Binani, James H Critchley, and Kurt S Anderson 181 Linear Dual Algebra Algorithms and their Application to Kinematics Ettore Pennestr`ı and Pier Paolo Valentini 207 A Memory Based Communication in the Co-simulation of Multibody and Finite Element Codes for Pantograph-Catenary Interaction Simulation Jorge Ambr´ osio, Jo˜ ao Pombo, Frederico Rauter, and Manuel Pereira 231 Physiological Methods to Solve the Force-Sharing Problem in Biomechanics Marko Ackermann and Werner Schiehlen Department of Biomedical Engineering, Cleveland Clinic Foundation, 9500 Euclid Avenue/ND20, 44195 Cleveland, OH, USA E-mail: ackermm2@ccf.org Institute of Engineering and Computational Mechanics, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany E-mail: schiehlen@itm.uni-stuttgart.de Summary The determination of individual muscle forces has many applications including the assessment of muscle coordination and internal loads on joints and bones, useful, for instance, for the design of endoprostheses Because muscle forces cannot be directly measured without invasive techniques, they are often estimated from joint moments by means of optimization procedures that search for a unique solution among the infinite solutions for the muscle forces that generate the same joint moments The conventional approach to solve this problem, the static optimization, is computationally efficient but neglects the dynamics involved in muscle force generation and requires the use of an instantaneous cost function, leading often to unrealistic estimations of muscle forces An alternative is using dynamic optimization associated with a motion tracking, which is, however, computationally very costly Other alternative approaches recently proposed in the literature are briefly reviewed and two new approaches are proposed to overcome the limitations of static optimization delivering more realistic estimations of muscle forces while being computationally less expensive than dynamic optimization Introduction Inverse dynamics is used to compute the net joint moments required to generate a measured motion Although giving a clue about the intensity of the actuation required to accomplish the observed motion, net joint moments fail in delivering information on the forces applied by the individual muscles and other structures spanning the joints Because the skeletal system is redundantly actuated by muscles, i.e there are many more muscles than actuated degrees of freedom, and many muscles are multi-articular, spanning more that one joint, the direct translation of net moments into muscle forces is not possible Therefore, conclusions about muscle activity from net joint moments are not very reliable (Zajac et al [33]) Furthermore, the energy consumption C.L Bottasso (ed.), Multibody Dynamics: Computational Methods and Applications, c Springer Science+Business Media B.V 2009 M Ackermann and W Schiehlen involved, represented by the metabolic cost during human motion cannot be accurately assessed In order to solve the mathematically indeterminate problem and assess muscle forces, optimization approaches are employed The classical static optimization approach is characterized by the search for muscle forces that minimize a cost function and fulfill constraints, given basically by bounded muscle forces and by the equations of motion or joint moments computed by inverse dynamics, respectively The cost functions are mathematical expressions assumed to model some physiological criteria optimized by the central nervous system during a particular activity In spite of being computationally efficient, the static optimization approach assumes an instantaneous optimal distribution of muscle forces suffering from two important limitations Firstly, it neglects the muscle contraction and activation dynamics, what might lead to unphysiological estimations of muscle forces Secondly, the cost functions must be an instantaneous measure of performance, what excludes the possibility of using time-integral criteria as for example total metabolic cost expended The latter limitation is specially important for the analysis of human walking, since metabolic cost is accepted to play an important role during locomotion The muscle activation and contraction dynamics can be taken into account by using dynamic optimization associated with the tracking of the prescribed kinematics This approach is based on the search for optimal controls, in this case the neural excitations, that drive a forward-dynamics model of the musculoskeletal system to track the prescribed motion Due to the several numerical integrations of the differential equations necessary, a prohibitive computational effort is required to achieve a solution This drawback prevents this approach from being widely used and stimulated recent efforts to reduce the computational burden Some strategies based on dynamic optimization are presented in more details in Section 3.2 An approach to solve the distribution problem in biomechanics is proposed in Section 4, see also Ackermann and Schiehlen [3] It considers the muscle contraction and activation dynamics and permits the use of time-integral cost functions as the total metabolic cost This approach is called extended inverse dynamics (EID) because it requires, in addition to the inversion of the skeletal system dynamics, the inversion of the muscle contraction and activation dynamics Since no numerical integration of the differential equations is required, the extended inverse dynamics is computationally less costly than the dynamic optimization A second, simplified approach, called modified static optimization (MSO), to permit computation of muscle forces that fulfill the constraints given by the activation and contraction dynamics is also proposed and presented in Section The latter approach maintains computational effort similar to the ones for static optimization, while considering the dynamics involved in the muscle force generation process ... on Multibody Dynamics was initiated in Lisbon in 2003, and then continued in Madrid in 2005 with the goal of providing researchers in Multibody Dynamics with appropriate venues for exchanging... carlo. bottasso@ polimi.it ISBN 97 8-1 -4 02 0-8 82 8-5 e-ISBN 97 8-1 -4 02 0-8 82 9-2 Library of Congress Control Number: 2008933572 All Rights Reserved c 2009 Springer Science + Business Media B.V No part of... Summary The determination of individual muscle forces has many applications including the assessment of muscle coordination and internal loads on joints and bones, useful, for instance, for the
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