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Research Papers

Multi-Integral Method for Solving the Forward Dynamics of Stiff Multibody Systems

[+] Author and Article Information
Paul Milenkovic

Department of Electrical and Computer Engineering,
University of Wisconsin-Madison,
Madison, WI 53706
e-mail: phmilenk@wisc.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received August 23, 2012; final manuscript received April 19, 2013; published online June 27, 2013. Assoc. Editor: Sergey Nersesov.

J. Dyn. Sys., Meas., Control 135(5), 051014 (Jun 27, 2013) (9 pages) Paper No: DS-12-1274; doi: 10.1115/1.4024300 History: Received August 23, 2012; Revised April 19, 2013

The Hermite–Obreshkov–Padé (HOP) procedure is an implicit method for the numerical solution of a system of ordinary differential equations (ODEs) applicable to stiff dynamical systems. This procedure applies an Obreshkov condition to multiple derivatives of the system state vector, both at the start and end of a time step in the numerical solution. That condition is shown to be satisfied by the Hermite interpolating polynomial that matches the state vector and its derivatives, also at the start and end of a time step. The Hermite polynomial, in turn, can be specified in terms of the system state and its derivatives at the start of a step together with a collection of free parameters. Adjusting these free parameters to minimize magnitudes of the ODE residual and its derivatives at the end of a step serves as a proxy for matching the system state and its derivatives. A high-order Taylor expansion at the start of a time step interval models the residual and its derivatives over the entire interval. A variant of this procedure adjusts those parameters to match integrals of the system state over the duration of that interval. This is done by minimizing magnitudes of integrals of the ODE residual calculated from the extrapolating Taylor-series expansion, a process that avoids the need to determine integration constants for multiple integrals of the state. This alternative method eliminates the calculation of high-order derivatives of the system state and hence avoids loss in accuracy from floating-point round off. Numerical performance is evaluated on a dynamically unbalanced constant-velocity (CV) coupling having a high spring rate constraining shaft deflection.

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