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Solving Differential-Algebraic Equation Systems: Alternative Index-2 and Index-1 Approaches for Constrained Mechanical Systems

[+] Author and Article Information
Bernhard Schweizer

Department of Mechanical Engineering,
Institute of Applied Dynamics,
Technical University Darmstadt,
Otto-Berndt-Strasse 2,
Darmstadt 64287, Germany
e-mail: schweizer@sds.tu-darmstadt.de

Pu Li

Department of Mechanical Engineering,
Institute of Applied Dynamics,
Technical University Darmstadt,
Otto-Berndt-Strasse 2,
Darmstadt 64287, Germany

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 23, 2014; final manuscript received August 6, 2015; published online November 13, 2015. Assoc. Editor: Dan Negrut.

J. Comput. Nonlinear Dynam 11(4), 044501 (Nov 13, 2015) (13 pages) Paper No: CND-14-1297; doi: 10.1115/1.4031287 History: Received November 23, 2014; Revised August 06, 2015

Regarding constrained mechanical systems, we are faced with index-3 differential-algebraic equation (DAE) systems. Direct discretization of the index-3 DAE systems only enforces the position constraints to be fulfilled at the integration-time points, but not the hidden constraints. In addition, order reduction effects are observed in the velocity variables and the Lagrange multipliers. In literature, different numerical techniques have been suggested to reduce the index of the system and to handle the numerical integration of constrained mechanical systems. This paper deals with an alternative concept, called collocated constraints approach. We present index-2 and index-1 formulations in combination with implicit Runge–Kutta methods. Compared with the direct discretization of the index-3 DAE system, the proposed method enforces also the constraints on velocity and—in case of the index-1 formulation—the constraints on acceleration level. The proposed method may very easily be implemented in standard Runge–Kutta solvers. Here, we only discuss mechanical systems. The presented approach can, however, also be applied for solving nonmechanical higher-index DAE systems.

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Figures

Grahic Jump Location
Fig. 1

Nonlinear pendulum

Grahic Jump Location
Fig. 2

Simulation results for the nonlinear pendulum: displacement y(t), velocity w(t), Lagrange multiplier λ(t), constraint g(t), hidden constraints g˙(t) and g¨(t) as well as total energy Etotal (Radau-IIa-S2-I3 (classical), Radau-IIa-S3-I3 (classical), Radau-IIa-S2-I2 (CCA), Radau-IIa-S3-I2 (CCA), and Radau-IIa-S3-I1 (CCA))

Grahic Jump Location
Fig. 3

Convergence plots for the nonlinear pendulum: global and local error over the step size h (Radau-IIa-S2-I3 (classical), Radau-IIa-S3-I3 (classical), Radau-IIa-S2-I2 (CCA), Radau-IIa-S3-I2 (CCA), and Radau-IIa-S3-I1 (CCA))

Grahic Jump Location
Fig. 4

Planar slider-crank mechanism

Grahic Jump Location
Fig. 5

Simulation results for the slider-crank mechanism: displacement xS1(t), velocity vS1(t), angle φ1(t), angular velocity ω1(t), Lagrange multiplier λin(t), constraint gin(t) and hidden constraints g˙in(t) and g¨in(t) (Radau-IIa-S2-I3 (classical), Radau-IIa-S3-I3 (classical), Radau-IIa-S2-I2 (CCA), Radau-IIa-S3-I2 (CCA), and Radau-IIa-S3-I1 (CCA)).

Grahic Jump Location
Fig. 6

Convergence plots for the slider-crank mechanism: global and local error over the stepsize h (Radau-IIa-S2-I3 (classical), Radau-IIa-S3-I3 (classical), Radau-IIa-S2-I2 (CCA), Radau-IIa-S3-I2 (CCA), and Radau-IIa-S3-I1 (CCA))

Grahic Jump Location
Fig. 7

Spherical four-bar mechanism

Grahic Jump Location
Fig. 8

Simulation results for the spherical four-bar mechanism: displacement xS2(t), velocity vS2(t), Bryant angle α2(t), angular velocity ω2,x(t), Lagrange multiplier λsp,x(t), constraint gsp,x(t) and hidden constraints g˙sp,x(t) and g¨sp,x(t) (Radau-IIa-S2-I3 (classical), Radau-IIa-S3-I3 (classical), Radau-IIa-S2-I2 (CCA), Radau-IIa-S3-I2 (CCA), and Radau-IIa-S3-I1 (CCA))

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