Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed

Hector Laos

doi:10.1007/978-3-030-75914-8_4

Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed

Hector Laos

Mechanical Systems Modeling

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

The equations of motion (EOM) for the heavy symmetrical top with one point fixed are highly nonlinear. The literature describes the numerical methods that are used to resolve this classical system, including modern tools, such as the Runge−Kutta fourth−order method. Finding the derivate of closed-form solutions for the EOM is more difficult and, as mentioned in the literature, discovering the solution is not always possible for all the EOM. Fortunately, a few examples are available that serve as a guide to move further in this topic. The purpose of this paper is to find a methodology that will produce the solutions for a given subset of EOMs that fulfill certain requisites. This paper summarizes the literature available on this topic and then follows with the derivation of the EOM using the Euler−Lagrange method. The Routhian method will be used to reduce the size of the expression, and it continues with the formulation of the classical cubic function, f(u), through a novel process. The roots of f(u) are of the utmost importance in finding the EOM closed-form solution, and once the final roots are selected, the general method that will produce the closed-form solutions is presented. Two sets of examples are included to show the validity of the process, and comparisons of the results from the closed-form solutions vs. the numerical results for these examples are shown.

Original language	English
Title of host publication	Special Topics in Structural Dynamics and Experimental Techniques, Volume 5 - Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics, 2021
Editors	David S. Epp
Publisher	Springer
Pages	29-38
Number of pages	10
ISBN (Print)	9783030759131
DOIs	https://doi.org/10.1007/978-3-030-75914-8_4
State	Published - 2022
Event	39th IMAC, A Conference and Exposition on Structural Dynamics, 2021 - Virtual, Online Duration: Feb 8 2021 → Feb 11 2021

Publication series

Name	Conference Proceedings of the Society for Experimental Mechanics Series
ISSN (Print)	2191-5644
ISSN (Electronic)	2191-5652

Conference

Conference	39th IMAC, A Conference and Exposition on Structural Dynamics, 2021
City	Virtual, Online
Period	02/8/21 → 02/11/21

Funding

This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http:// energy.gov/downloads/doe-public-access-plan).

Keywords

Closed-form solutions
Cubic polynomial f(u)
Equations of motion
Gyroscopes
Routhian

Access to Document

10.1007/978-3-030-75914-8_4

Cite this

Laos, H. (2022). Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed. In D. S. Epp (Ed.), Special Topics in Structural Dynamics and Experimental Techniques, Volume 5 - Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics, 2021 (pp. 29-38). (Conference Proceedings of the Society for Experimental Mechanics Series). Springer. https://doi.org/10.1007/978-3-030-75914-8_4

Laos, Hector. / Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed. Special Topics in Structural Dynamics and Experimental Techniques, Volume 5 - Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics, 2021. editor / David S. Epp. Springer, 2022. pp. 29-38 (Conference Proceedings of the Society for Experimental Mechanics Series).

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title = "Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed",

abstract = "The equations of motion (EOM) for the heavy symmetrical top with one point fixed are highly nonlinear. The literature describes the numerical methods that are used to resolve this classical system, including modern tools, such as the Runge−Kutta fourth−order method. Finding the derivate of closed-form solutions for the EOM is more difficult and, as mentioned in the literature, discovering the solution is not always possible for all the EOM. Fortunately, a few examples are available that serve as a guide to move further in this topic. The purpose of this paper is to find a methodology that will produce the solutions for a given subset of EOMs that fulfill certain requisites. This paper summarizes the literature available on this topic and then follows with the derivation of the EOM using the Euler−Lagrange method. The Routhian method will be used to reduce the size of the expression, and it continues with the formulation of the classical cubic function, f(u), through a novel process. The roots of f(u) are of the utmost importance in finding the EOM closed-form solution, and once the final roots are selected, the general method that will produce the closed-form solutions is presented. Two sets of examples are included to show the validity of the process, and comparisons of the results from the closed-form solutions vs. the numerical results for these examples are shown.",

keywords = "Closed-form solutions, Cubic polynomial f(u), Equations of motion, Gyroscopes, Routhian",

author = "Hector Laos",

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Laos, H 2022, Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed. in DS Epp (ed.), Special Topics in Structural Dynamics and Experimental Techniques, Volume 5 - Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics, 2021. Conference Proceedings of the Society for Experimental Mechanics Series, Springer, pp. 29-38, 39th IMAC, A Conference and Exposition on Structural Dynamics, 2021, Virtual, Online, 02/8/21. https://doi.org/10.1007/978-3-030-75914-8_4

Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed. / Laos, Hector.
Special Topics in Structural Dynamics and Experimental Techniques, Volume 5 - Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics, 2021. ed. / David S. Epp. Springer, 2022. p. 29-38 (Conference Proceedings of the Society for Experimental Mechanics Series).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N2 - The equations of motion (EOM) for the heavy symmetrical top with one point fixed are highly nonlinear. The literature describes the numerical methods that are used to resolve this classical system, including modern tools, such as the Runge−Kutta fourth−order method. Finding the derivate of closed-form solutions for the EOM is more difficult and, as mentioned in the literature, discovering the solution is not always possible for all the EOM. Fortunately, a few examples are available that serve as a guide to move further in this topic. The purpose of this paper is to find a methodology that will produce the solutions for a given subset of EOMs that fulfill certain requisites. This paper summarizes the literature available on this topic and then follows with the derivation of the EOM using the Euler−Lagrange method. The Routhian method will be used to reduce the size of the expression, and it continues with the formulation of the classical cubic function, f(u), through a novel process. The roots of f(u) are of the utmost importance in finding the EOM closed-form solution, and once the final roots are selected, the general method that will produce the closed-form solutions is presented. Two sets of examples are included to show the validity of the process, and comparisons of the results from the closed-form solutions vs. the numerical results for these examples are shown.

AB - The equations of motion (EOM) for the heavy symmetrical top with one point fixed are highly nonlinear. The literature describes the numerical methods that are used to resolve this classical system, including modern tools, such as the Runge−Kutta fourth−order method. Finding the derivate of closed-form solutions for the EOM is more difficult and, as mentioned in the literature, discovering the solution is not always possible for all the EOM. Fortunately, a few examples are available that serve as a guide to move further in this topic. The purpose of this paper is to find a methodology that will produce the solutions for a given subset of EOMs that fulfill certain requisites. This paper summarizes the literature available on this topic and then follows with the derivation of the EOM using the Euler−Lagrange method. The Routhian method will be used to reduce the size of the expression, and it continues with the formulation of the classical cubic function, f(u), through a novel process. The roots of f(u) are of the utmost importance in finding the EOM closed-form solution, and once the final roots are selected, the general method that will produce the closed-form solutions is presented. Two sets of examples are included to show the validity of the process, and comparisons of the results from the closed-form solutions vs. the numerical results for these examples are shown.

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Laos H. Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed. In Epp DS, editor, Special Topics in Structural Dynamics and Experimental Techniques, Volume 5 - Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics, 2021. Springer. 2022. p. 29-38. (Conference Proceedings of the Society for Experimental Mechanics Series). doi: 10.1007/978-3-030-75914-8_4

Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed

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