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Research Papers

Determination of Holonomic and Nonholonomic Constraint Reactions in an Index-3 Augmented Lagrangian Formulation With Velocity and Acceleration Projections

[+] Author and Article Information
Daniel Dopico

Department of Computer Science and
Department of Mechanical Engineering,
Virginia Tech,
2200 Kraft Drive,
Blacksburg, VA 24060
e-mail: ddopico@udc.es

Francisco González

Department of Mechanical Engineering
and Centre for Intelligent Machines,
McGill University,
817 Sherbrooke Street West,
Montréal, QC H3A 0C3, Canada
e-mail: franglez@cim.mcgill.ca

Javier Cuadrado

Laboratorio de Ingeniería Mecánica,
University of La Coruña,
Mendizábal s/n,
Ferrol 15403, Spain
e-mail: javicuad@cdf.udc.es

József Kövecses

Department of Mechanical Engineering and Centre for Intelligent Machines,
McGill University,
817 Sherbrooke Street West,
Montréal, QC H3A 0C3, Canada
e-mail: jozsef.kovecses@mcgill.ca

1Corresponding author.

Manuscript received July 20, 2013; final manuscript received May 9, 2014; published online July 11, 2014. Assoc. Editor: Dan Negrut.

J. Comput. Nonlinear Dynam 9(4), 041006 (Jul 11, 2014) (9 pages) Paper No: CND-13-1185; doi: 10.1115/1.4027671 History: Received July 20, 2013; Revised May 09, 2014

Index-3 augmented Lagrangian formulations with projections of velocities and accelerations represent an efficient and robust method to carry out the forward-dynamics simulation of multibody systems modeled in dependent coordinates. Existing formalisms, however, were only established for holonomic systems, for which the expression of the constraints at the position-level is known. In this work, an extension of the original algorithms for nonholonomic systems is introduced. Moreover, projections of velocities and accelerations have two side effects: they modify the kinetic energy of the system and they contribute to the constraint reaction forces. Although the effects of the projections on the energy have been studied by several authors, their role in the calculation of the reaction forces has not been described so far. In this work, expressions to determine the constraint reactions from the Lagrange multipliers of the dynamic equations and the Lagrange multipliers of the velocity and acceleration projections are introduced. Simulation results show that the proposed strategy can be used to expand the capabilities of index-3 augmented Lagrangian algorithms, making them able to deal with nonholonomic constraints and provide correct reaction efforts.

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References

Figures

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Fig.4

Reaction fy at contact point C during 2D motion of the disk, obtained with the projection of velocities (fy vel) and accelerations (fy acc)

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Fig. 1

Nonholonomic constraint restraining the velocity of mass A to be collinear with segment A–B

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Fig. 5

Reaction forces fx and fy at contact point C during motion of the disk, compared to mx··P and my··P

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Fig. 7

Actuation forces: (a) Bayo's (top) and (b) evolved Cuadrado's (bottom)

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Fig. 2

Reaction forces acting on mass A in the x and y directions due to the nonholonomic constraint in the first example, computed with direct solution (54) and with the ALI3-P formulation

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Fig. 3

3D motion of a disk rolling on a plane

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