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On the constraints violation in forward dynamics of multibody systems

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Abstract

It is known that the dynamic equations of motion for constrained mechanical multibody systems are frequently formulated using the Newton–Euler’s approach, which is augmented with the acceleration constraint equations. This formulation results in the establishment of a mixed set of partial differential and algebraic equations, which are solved in order to predict the dynamic behavior of general multibody systems. The classical solution of the equations of motion is highly prone to constraints violation because the position and velocity constraint equations are not fulfilled. In this work, a general and comprehensive methodology to eliminate the constraints violation at the position and velocity levels is offered. The basic idea of the described approach is to add corrective terms to the position and velocity vectors with the intent to satisfy the corresponding kinematic constraint equations. These corrective terms are evaluated as a function of the Moore–Penrose generalized inverse of the Jacobian matrix and of the kinematic constraint equations. The described methodology is embedded in the standard method to solve the equations of motion based on the technique of Lagrange multipliers. Finally, the effectiveness of the described methodology is demonstrated through the dynamic modeling and simulation of different planar and spatial multibody systems. The outcomes in terms of constraints violation at the position and velocity levels, conservation of the total energy and computational efficiency are analyzed and compared with those obtained with the standard Lagrange multipliers method, the Baumgarte stabilization method, the augmented Lagrangian formulation, the index-1 augmented Lagrangian, and the coordinate partitioning method.

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Acknowledgements

The first author expresses his gratitude to the Portuguese Foundation for Science and Technology through the PhD grant (PD/BD/114154/2016). This work has been supported by the Portuguese Foundation for Science and Technology with the reference project UID/EEA/04436/2013, by FEDER funds through the COMPETE 2020—Programa Operacional Competitividade e Internacionalização (POCI) with the reference project POCI-01-0145-FEDER-006941. The authors also would like to acknowledge the considerable contributions of Professor Javier Cuadrado from University of A Coruña, Spain, for sharing with us some thoughts and material for the numerical examples of application. Finally, the authors are much indebted to the anonymous reviewers for useful comments, recommendations and suggestions.

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Minho, Campus de Azurém, 4804-533, Guimarães, Portugal

    Filipe Marques & Paulo Flores

  2. Department of Textile Engineering, University of Minho, Campus de Azurém, 4804-533, Guimarães, Portugal

    António P. Souto

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  1. Filipe Marques
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Correspondence to Paulo Flores.

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Marques, F., Souto, A.P. & Flores, P. On the constraints violation in forward dynamics of multibody systems. Multibody Syst Dyn 39, 385–419 (2017). https://doi.org/10.1007/s11044-016-9530-y

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