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Research Papers

A Parametric Study on the Baumgarte Stabilization Method for Forward Dynamics of Constrained Multibody Systems

[+] Author and Article Information
Paulo Flores1

Departamento de Engenharia Mecânica, Universidade do Minho, Campus de Azurém, 4800-058 Guimarães, Portugalpflores@dem.uminho.pt

Margarida Machado

Departamento de Engenharia Mecânica, Universidade do Minho, Campus de Azurém, 4800-058 Guimarães, Portugalmargarida@dem.uminho.pt

Eurico Seabra

Departamento de Engenharia Mecânica, Universidade do Minho, Campus de Azurém, 4800-058 Guimarães, Portugaleseabra@dem.uminho.pt

Miguel Tavares da Silva

Instituto de Engenharia Mecânica (IDMEC), Instituto Superior Técnico, Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugalmiguelsilva@ist.utl.pt

1

Corresponding author.

J. Comput. Nonlinear Dynam 6(1), 011019 (Oct 13, 2010) (9 pages) doi:10.1115/1.4002338 History: Received July 30, 2009; Revised December 10, 2009; Published October 13, 2010; Online October 13, 2010

This paper presents and discusses the results obtained from a parametric study on the Baumgarte stabilization method for forward dynamics of constrained multibody systems. The main purpose of this work is to analyze the influence of the variables that affect the violation of constraints, chiefly the values of the Baumgarte parameters, the integration method, the time step, and the quality of the initial conditions for the positions. In the sequel of this process, the formulation of the rigid multibody systems is reviewed. The generalized Cartesian coordinates are selected as the variables to describe the bodies’ degrees of freedom. The formulation of the equations of motion uses the Newton–Euler approach, augmented with the constraint equations that lead to a set of differential algebraic equations. Furthermore, the main issues related to the stabilization of the violation of constraints based on the Baumgarte approach are revised. Special attention is also given to some techniques that help in the selection process of the values of the Baumgarte parameters, namely, those based on the Taylor’s series and the Laplace transform technique. Finally, a slider-crank mechanism with eccentricity is considered as an example of application in order to illustrate how the violation of constraints can be affected by different factors.

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Figures

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Figure 1

Open-loop and closed-loop control systems

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Figure 2

Behavior of Eq. 21 for different values of α and β

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Figure 3

Behavior of Eq. 28 for different values of α

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Figure 4

Behavior of Eq. 34 for different values of α and β

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Figure 5

Mapping regions of the s-plane into the z-plane

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Figure 6

Stability region in the α̱-β̱ plane for the Euler method

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Figure 7

Model of the eccentric slider-crank mechanism

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Figure 8

Error of the slider position and third constraint equation obtained with and without constraints violation stabilization: (a) slider position error and (b) error of the third constraint equation

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Figure 9

Influence of the values of the Baumgarte parameters on the error of the third constraint equation and its derivative: (a) error of the third constraint equation, (b) error of the third constraint equation, (c) error of the derivative of the third constraint equation, and (d) error of the derivative of the third constraint equation

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Figure 10

Influence of the integrator used on the error of the third constraint equation and its derivative: (a) error of the third constraint equation, (b) error of the third constraint equation, (c) error of the derivative of the third constraint equation, and (d) error of the derivative of the third constraint equation

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Figure 11

Influence of the time step size used on the error of the third constraint equation and its derivative: (a) error of the third constraint equation, (b) error of the third constraint equation, (c) error of the derivative of the third constraint equation, and (d) error of the derivative of the third constraint equation

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Figure 12

Influence of the initial conditions on the error of the third constraint equation and its derivative: (a) error of the third constraint equation and (b) error of the derivative of the third constraint equation

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