Research Papers

Scaling of Constraints and Augmented Lagrangian Formulations in Multibody Dynamics Simulations

[+] Author and Article Information
Olivier A. Bauchau1

Daniel Guggenheim School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332olivier.bauchau@ae.gatech.edu

Alexander Epple

Daniel Guggenheim School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332alexander.epple@gatech.edu

Carlo L. Bottasso

Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Milano 20156, Italycarlo.bottasso@polimi.it


Corresponding author.

J. Comput. Nonlinear Dynam 4(2), 021007 (Mar 09, 2009) (9 pages) doi:10.1115/1.3079826 History: Received November 30, 2007; Revised August 04, 2008; Published March 09, 2009

This paper addresses practical issues associated with the numerical enforcement of constraints in flexible multibody systems, which are characterized by index-3 differential algebraic equations (DAEs). The need to scale the equations of motion is emphasized; in the proposed approach, they are scaled based on simple physical arguments, and an augmented Lagrangian term is added to the formulation. Time discretization followed by a linearization of the resulting equations leads to a Jacobian matrix that is independent of the time step size, h; hence, the condition number of the Jacobian and error propagation are both O(h0): the numerical solution of index-3 DAEs behaves as in the case of regular ordinary differential equations (ODEs). Since the scaling factor depends on the physical properties of the system, the proposed scaling decreases the dependency of this Jacobian on physical properties, further improving the numerical conditioning of the resulting linearized equations. Because the scaling of the equations is performed before the time and space discretizations, its benefits are reaped for all time integration schemes. The augmented Lagrangian term is shown to be indispensable if the solution of the linearized system of equations is to be performed without pivoting, a requirement for the efficient solution of the sparse system of linear equations. Finally, a number of numerical examples demonstrate the efficiency of the proposed approach to scaling.

Copyright © 2009 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 2

Beam actuated by a tip crank

Grahic Jump Location
Figure 3

Displacement components at the beam’s midspan: u1: solid line, u2: dashed line, and u3: dashed-dotted line

Grahic Jump Location
Figure 4

Convergence characteristics of three integration schemes: Radau IIA: solid line, energy decaying scheme: dashed-dotted line, and HHT: dashed line




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