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Research Papers

Efficient Solution of Maggi’s Equations

[+] Author and Article Information
Javier García de Jalón

ETSII and INSIA,  Technical University of Madrid (UPM), José Gutiérrez Abascal 2, 28006 Madrid, Spainjavier.garciadejalon@upm.es INSIA, Technical University of Madrid (UPM), Campus Sur UPM, Ctra. Valencia km 7, 28031 Madrid, Spainjavier.garciadejalon@upm.es

Alfonso Callejo

ETSII and INSIA,  Technical University of Madrid (UPM), José Gutiérrez Abascal 2, 28006 Madrid, Spaina.callejo@upm.es INSIA, Technical University of Madrid (UPM), Campus Sur UPM, Ctra. Valencia km 7, 28031 Madrid, Spaina.callejo@upm.es

Andrés F. Hidalgo

ETSII and INSIA,  Technical University of Madrid (UPM), José Gutiérrez Abascal 2, 28006 Madrid, Spainandres.francisco.hidalgo@upm.es INSIA, Technical University of Madrid (UPM), Campus Sur UPM, Ctra. Valencia km 7, 28031 Madrid, Spainandres.francisco.hidalgo@upm.es

J. Comput. Nonlinear Dynam 7(2), 021003 (Dec 22, 2011) (10 pages) doi:10.1115/1.4005238 History: Received May 16, 2011; Revised October 03, 2011; Published December 22, 2011; Online December 22, 2011

According to a recent paper (Laulusa and Bauchau, 2008, “Review of Classical Approaches for Constraint Enforcement in Multibody Systems,” ASME J. Comput. Nonlinear Dyn., 3 (1), 011004), Maggi’s formulation is a simple and stable way to solve the dynamic equations of constrained multibody systems. Among the difficulties of Maggi’s formulation, Laulusa and Bauchau quoted the need for an appropriate choice (and change, when necessary) of independent coordinates, as well as the high cost of computing and updating the basis of the tangent null space of constraint equations. In this paper, index-1 Lagrange’s equations are first considered, including the not-so-rare case of having a singular mass matrix and redundant constraints. The existence and uniqueness of solution for acceleration vector and Lagrange multipliers vector is studied in a very simple way. Then, following Von Schwerin (Von Schwerin, Multibody System Simulation. Numerical Methods, Algorithms and Software, Springer, New York, 1999), Maggi’s formulation is described as the most efficient way (in general) to solve these index-1 equations. Next, an improved double-step method, which implements the matrix transformations of Maggi’s formulation in an efficient way, is described. Finally, two large real-life examples are presented.

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Copyright © 2012 by American Society of Mechanical Engineers
Topics: Equations , Chain
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Figures

Grahic Jump Location
Figure 3

Example of an open-chain system

Grahic Jump Location
Figure 4

Loop closure by means of a revolute joint (a) or a rod element (b)

Grahic Jump Location
Figure 5

Schematic MBS model of the coach

Grahic Jump Location
Figure 6

Schematic MBS model of the truck

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