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Research Papers

A Comparison of Ordinary Differential Equation Solvers for Dynamical Systems With Impacts

[+] Author and Article Information
Spyridon Dallas

Department of Mechanical Engineering,
National Technical University of Athens,
9 Heroon Polytechniou Street,
Athens 15780, Greece
e-mail: spyro.d.mechs@gmail.com

Konstantinos Machairas

Department of Mechanical Engineering,
National Technical University of Athens,
9 Heroon Polytechniou Street,
Athens 15780, Greece
e-mail: kmach@central.ntua.gr

Evangelos Papadopoulos

Mem. ASME
Department of Mechanical Engineering,
National Technical University of Athens,
9 Heroon Polytechniou Street,
Athens 15780, Greece
e-mail: egpapado@central.ntua.gr

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 4, 2016; final manuscript received May 24, 2017; published online September 7, 2017. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 12(6), 061016 (Sep 07, 2017) (8 pages) Paper No: CND-16-1600; doi: 10.1115/1.4037074 History: Received December 04, 2016; Revised May 24, 2017

In this paper, a method is developed that results in guidelines for selecting the best Ordinary Differential Equation (ODE) solver and its parameters, for a class of nonlinear hybrid system were impacts are present. A monopod interacting compliantly with the ground is introduced as a new benchmark problem, and is used to compare the various solvers available in the widely used matlab ode suite. To provide result generality, the mathematical description of the hybrid system is brought to a dimensionless form, and its dimensionless parameters are selected in a range taken from existing systems and corresponding to different levels of numerical stiffness. The effect of error tolerance and phase transition strategy is taken into account. The obtained system responses are evaluated using solution speed and accuracy criteria. It is shown that hybrid systems represent a class of problems that cycle between phases in which the system of the equations of motion (EoM) is stiff (interaction with the ground), and phases in which it is not (flight phases); for such systems, the appropriate type of solver was an open question. Based on this evaluation, both general and case-specific guidelines are provided for selecting the most appropriate ODE solver. Interestingly, the best solver for a realistic test case turned out to be a solver recommended for numerically nonstiff ODE problems.

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Figures

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Fig. 1

Two snapshots of a vertically hopping monopod at different phases: (a) flight phase and (b) stance phase

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Fig. 2

Response of hopping monopod by various solvers, for error tolerance (1 × 10−3) and transition by “if…else…end” statement

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Fig. 3

Absolute error between the reference toe displacement and the toe displacement provided by an ODE solver

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Fig. 4

Correlation of absolute and relative error from reference with DT and IAE criteria

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