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Research Papers

Integration of Flexible Multibody Systems Using Radau IIA Algorithms

[+] Author and Article Information
Jielong Wang1

 Gamma Technologies, Inc., 601 Oakmont Lane Suite 220, Westmont, IL 60559j.wang@gtisoft.com

Jesus Rodriguez, Rifat Keribar

 Gamma Technologies, Inc., 601 Oakmont Lane Suite 220, Westmont, IL 60559

1

Corresponding author.

J. Comput. Nonlinear Dynam 5(4), 041008 (Jul 28, 2010) (14 pages) doi:10.1115/1.4001907 History: Received March 02, 2009; Revised November 06, 2009; Published July 28, 2010; Online July 28, 2010

The two-stage and three-stage Radau IIA stiff integrators, belonging to the implicit Runge–Kutta family, are implemented in a computationally efficient manner to solve flexible multibody systems. These problems feature large displacements and finite rotations together with small elastic deformations. The two-stage and three-stage algorithms are modified to integrate the finite element based, second order differential equations of multibody dynamics directly. A new simplified Newton iteration is implemented for the two-stage algorithm to reduce its computational cost. The resulting linear system of equations obtained at each Newton iteration is solved efficiently for both the two-stage and three-stage algorithms. The Jacobian matrix in the simplified Newton iterations is evaluated through numerical differentiation. A compact storage strategy is used to archive the banded, sparse matrices. The error estimation based on the embedded formula and step size selection are discussed in detail. The proposed schemes are validated with the help of different numerical examples. The simulation results are consistent with those of other types of integrators. These examples show that Radau IIA algorithms can solve stiff problems with acceptable speed while guaranteeing stability and accuracy.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 10

In-line six cylinder crankshaft: time history of the external loads applied to a given piston

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Figure 11

In-line six cylinder crankshaft: time history of the spring reaction force computed by HHT algorithm (◇), two-stage scheme (◁), and three-stage scheme (▷)

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Figure 1

Spring-mass system: error in mass displacement versus number of time steps computed by HHT algorithm (◇), two-stage scheme (◁), and three-stage scheme (○)

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Figure 2

Crank-slider mechanism

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Figure 3

Flexible crank-slider: time history of the slider displacement computed by Cash–Karp embedded formula (◁), HHT algorithm (▷), two-stage scheme (◇), and three-stage scheme (○)

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Figure 4

Flexible crank-slider: time history of the constraint torque computed by Cash–Karp embedded formula (◁), HHT algorithm (▷), two-stage scheme (◇), and three-stage scheme (○)

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Figure 5

Flexible crank-slider: error in slider velocity versus number of time steps computed by HHT algorithm (◇), two-stage scheme (◁), and three-stage scheme (○)

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Figure 6

Flexible crank-slider: error in crank bending moment versus number of time steps computed by HHT algorithm (◇), two-stage scheme (◁), and three-stage scheme (○)

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Figure 7

Flexible crank-slider: error in slider velocity versus CPU time computed by HHT algorithm (◇), two-stage scheme (◁), and three-stage scheme (○)

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Figure 8

In-line six cylinder crankshaft

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Figure 9

In-line cylinder crank-slider mechanism

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Figure 12

In-line six cylinder crankshaft: error in piston velocity versus number of time steps computed by HHT algorithm (◇), two-stage scheme (◁), and three-stage scheme (○)

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Figure 13

In-line six cylinder crankshaft: error in spring reaction versus number of time steps computed by HHT algorithm (◇), two-stage scheme (◁), and three-stage scheme (○)

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Figure 14

In-line six cylinder crankshaft: error in piston velocity versus CPU time computed by HHT algorithm (◇), two-stage scheme (◁), and three-stage scheme (○)

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