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Open AccessFeature PaperArticle
A Family of Conditionally Explicit Methods for Second-Order ODEs and DAEs: Application in Multibody Dynamics
by
Igor Fernández de Bustos
Igor Fernández de Bustos 1,*,
Haritz Uriarte
Haritz Uriarte 2,
Gorka Urkullu
Gorka Urkullu 3 and
Ibai Coria
Ibai Coria 3
1
Department of Mechanical Engineering, Faculty of Engineering of Bilbao, University of the Basque Country, Alameda Urquijo s/n, 48013 Bilbao, Spain
2
Department of Graphic Design and Engineering Projects, Faculty of Engineering of Bilbao, University of the Basque Country, Alameda Urquijo s/n, 48013 Bilbao, Spain
3
Department of Applied Mathematics, Faculty of Engineering of Bilbao, University of the Basque Country, Alameda Urquijo s/n, 48013 Bilbao, Spain
*
Author to whom correspondence should be addressed.
Mathematics 2024, 12(18), 2862; https://doi.org/10.3390/math12182862 (registering DOI)
Submission received: 29 July 2024
/
Revised: 5 September 2024
/
Accepted: 12 September 2024
/
Published: 14 September 2024
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.
Share and Cite
MDPI and ACS Style
Fernández de Bustos, I.; Uriarte, H.; Urkullu, G.; Coria, I.
A Family of Conditionally Explicit Methods for Second-Order ODEs and DAEs: Application in Multibody Dynamics. Mathematics 2024, 12, 2862.
https://doi.org/10.3390/math12182862
AMA Style
Fernández de Bustos I, Uriarte H, Urkullu G, Coria I.
A Family of Conditionally Explicit Methods for Second-Order ODEs and DAEs: Application in Multibody Dynamics. Mathematics. 2024; 12(18):2862.
https://doi.org/10.3390/math12182862
Chicago/Turabian Style
Fernández de Bustos, Igor, Haritz Uriarte, Gorka Urkullu, and Ibai Coria.
2024. "A Family of Conditionally Explicit Methods for Second-Order ODEs and DAEs: Application in Multibody Dynamics" Mathematics 12, no. 18: 2862.
https://doi.org/10.3390/math12182862
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