A Family of Conditionally Explicit Methods for Second-Order ODEs and DAEs: Application in Multibody Dynamics

Igor Fernández de Bustos (), Haritz Uriarte, Gorka Urkullu and Ibai Coria
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Igor Fernández de Bustos: Department of Mechanical Engineering, Faculty of Engineering of Bilbao, University of the Basque Country, Alameda Urquijo s/n, 48013 Bilbao, Spain
Haritz Uriarte: Department of Graphic Design and Engineering Projects, Faculty of Engineering of Bilbao, University of the Basque Country, Alameda Urquijo s/n, 48013 Bilbao, Spain
Gorka Urkullu: Department of Applied Mathematics, Faculty of Engineering of Bilbao, University of the Basque Country, Alameda Urquijo s/n, 48013 Bilbao, Spain
Ibai Coria: Department of Applied Mathematics, Faculty of Engineering of Bilbao, University of the Basque Country, Alameda Urquijo s/n, 48013 Bilbao, Spain

Mathematics, 2024, vol. 12, issue 18, 1-28

Abstract: There are several common procedures used to numerically integrate second-order ordinary differential equations. The most common one is to reduce the equation’s order by duplicating the number of variables. This allows one to take advantage of the family of Runge–Kutta methods or the Adams family of multi-step methods. Another approach is the use of methods that have been developed to directly integrate an ordinary differential equation without increasing the number of variables. An important drawback when using Runge–Kutta methods is that when one tries to apply them to differential algebraic equations, they require a reduction in the index, leading to a need for stabilization methods to remove the drift. In this paper, a new family of methods for the direct integration of second-order ordinary differential equations is presented. These methods can be considered as a generalization of the central differences method. The methods are classified according to the number of derivatives they take into account (degree). They include some parameters that can be chosen to configure the equation’s behavior. Some sets of parameters were studied, and some examples belonging to structural dynamics and multibody dynamics are presented. An example of the application of the method to a differential algebraic equation is also included.

Keywords: ordinary differential equations; differential algebraic equations; multibody dynamics; structural dynamics (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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