Simple but accurate periodic solutions for the nonlinear pendulum equation
Autor(a) principal: | |
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Data de Publicação: | 2019 |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UnB |
Texto Completo: | https://repositorio.unb.br/handle/10482/36547 https://doi.org/10.1590/1806-9126-rbef-2018-0202 http://orcid.org/0000-0001-5884-6621 |
Resumo: | Despite its elementary structure, the simple pendulum oscillations are described by a nonlinear differential equation whose exact solution for the angular displacement from vertical as a function of time cannot be expressed in terms of an elementary function, so either a numerical treatment or some analytical approximation is ultimately demanded. Such solutions have been thoroughly investigated due to the abundance of distinct pendular systems in nature and, more recently, due to the availability of automatic data acquisition systems in undergraduate laboratories. However, it is well-known that numerical solutions to differential equations usually loose accuracy (due to accumulation of roundoff errors) and polynomial approximations diverge after long time intervals. In this work, I take a few terms of the Fourier series expansion of the elliptic function sn ( u ; k ) as a source of accurate periodic solutions for the pendulum equation. Interestingly, these approximations remain accurate for arbitrarily long time intervals, even for large amplitudes, which shows its adequacy for the analysis of experimental data gathered in classical mechanics classes. |
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Simple but accurate periodic solutions for the nonlinear pendulum equationPênduloVibraçãoFourier, Séries deFunções elípticasDespite its elementary structure, the simple pendulum oscillations are described by a nonlinear differential equation whose exact solution for the angular displacement from vertical as a function of time cannot be expressed in terms of an elementary function, so either a numerical treatment or some analytical approximation is ultimately demanded. Such solutions have been thoroughly investigated due to the abundance of distinct pendular systems in nature and, more recently, due to the availability of automatic data acquisition systems in undergraduate laboratories. However, it is well-known that numerical solutions to differential equations usually loose accuracy (due to accumulation of roundoff errors) and polynomial approximations diverge after long time intervals. In this work, I take a few terms of the Fourier series expansion of the elliptic function sn ( u ; k ) as a source of accurate periodic solutions for the pendulum equation. Interestingly, these approximations remain accurate for arbitrarily long time intervals, even for large amplitudes, which shows its adequacy for the analysis of experimental data gathered in classical mechanics classes.Sociedade Brasileira de Física2020-01-24T10:31:44Z2020-01-24T10:31:44Z2019info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfLima, Fábio Menezes de Souza. Simple but accurate periodic solutions for the nonlinear pendulum equation. Revista Brasileira de Ensino de Física, v. 41, n. 1, e20180202, 2019. DOI: https://doi.org/10.1590/1806-9126-rbef-2018-0202. Disponível em: http://scielo.br/scielo.php?script=sci_arttext&pid=S1806-11172019000100413. Acesso em: 23 jan. 2020.https://repositorio.unb.br/handle/10482/36547https://doi.org/10.1590/1806-9126-rbef-2018-0202http://orcid.org/0000-0001-5884-6621(CC BY) - LIcença Creative Commons.info:eu-repo/semantics/openAccessLima, Fábio Menezes de Souzaengreponame:Repositório Institucional da UnBinstname:Universidade de Brasília (UnB)instacron:UNB2023-05-27T00:19:55Zoai:repositorio.unb.br:10482/36547Repositório InstitucionalPUBhttps://repositorio.unb.br/oai/requestrepositorio@unb.bropendoar:2023-05-27T00:19:55Repositório Institucional da UnB - Universidade de Brasília (UnB)false |
dc.title.none.fl_str_mv |
Simple but accurate periodic solutions for the nonlinear pendulum equation |
title |
Simple but accurate periodic solutions for the nonlinear pendulum equation |
spellingShingle |
Simple but accurate periodic solutions for the nonlinear pendulum equation Lima, Fábio Menezes de Souza Pêndulo Vibração Fourier, Séries de Funções elípticas |
title_short |
Simple but accurate periodic solutions for the nonlinear pendulum equation |
title_full |
Simple but accurate periodic solutions for the nonlinear pendulum equation |
title_fullStr |
Simple but accurate periodic solutions for the nonlinear pendulum equation |
title_full_unstemmed |
Simple but accurate periodic solutions for the nonlinear pendulum equation |
title_sort |
Simple but accurate periodic solutions for the nonlinear pendulum equation |
author |
Lima, Fábio Menezes de Souza |
author_facet |
Lima, Fábio Menezes de Souza |
author_role |
author |
dc.contributor.author.fl_str_mv |
Lima, Fábio Menezes de Souza |
dc.subject.por.fl_str_mv |
Pêndulo Vibração Fourier, Séries de Funções elípticas |
topic |
Pêndulo Vibração Fourier, Séries de Funções elípticas |
description |
Despite its elementary structure, the simple pendulum oscillations are described by a nonlinear differential equation whose exact solution for the angular displacement from vertical as a function of time cannot be expressed in terms of an elementary function, so either a numerical treatment or some analytical approximation is ultimately demanded. Such solutions have been thoroughly investigated due to the abundance of distinct pendular systems in nature and, more recently, due to the availability of automatic data acquisition systems in undergraduate laboratories. However, it is well-known that numerical solutions to differential equations usually loose accuracy (due to accumulation of roundoff errors) and polynomial approximations diverge after long time intervals. In this work, I take a few terms of the Fourier series expansion of the elliptic function sn ( u ; k ) as a source of accurate periodic solutions for the pendulum equation. Interestingly, these approximations remain accurate for arbitrarily long time intervals, even for large amplitudes, which shows its adequacy for the analysis of experimental data gathered in classical mechanics classes. |
publishDate |
2019 |
dc.date.none.fl_str_mv |
2019 2020-01-24T10:31:44Z 2020-01-24T10:31:44Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
Lima, Fábio Menezes de Souza. Simple but accurate periodic solutions for the nonlinear pendulum equation. Revista Brasileira de Ensino de Física, v. 41, n. 1, e20180202, 2019. DOI: https://doi.org/10.1590/1806-9126-rbef-2018-0202. Disponível em: http://scielo.br/scielo.php?script=sci_arttext&pid=S1806-11172019000100413. Acesso em: 23 jan. 2020. https://repositorio.unb.br/handle/10482/36547 https://doi.org/10.1590/1806-9126-rbef-2018-0202 http://orcid.org/0000-0001-5884-6621 |
identifier_str_mv |
Lima, Fábio Menezes de Souza. Simple but accurate periodic solutions for the nonlinear pendulum equation. Revista Brasileira de Ensino de Física, v. 41, n. 1, e20180202, 2019. DOI: https://doi.org/10.1590/1806-9126-rbef-2018-0202. Disponível em: http://scielo.br/scielo.php?script=sci_arttext&pid=S1806-11172019000100413. Acesso em: 23 jan. 2020. |
url |
https://repositorio.unb.br/handle/10482/36547 https://doi.org/10.1590/1806-9126-rbef-2018-0202 http://orcid.org/0000-0001-5884-6621 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.rights.driver.fl_str_mv |
(CC BY) - LIcença Creative Commons. info:eu-repo/semantics/openAccess |
rights_invalid_str_mv |
(CC BY) - LIcença Creative Commons. |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.publisher.none.fl_str_mv |
Sociedade Brasileira de Física |
publisher.none.fl_str_mv |
Sociedade Brasileira de Física |
dc.source.none.fl_str_mv |
reponame:Repositório Institucional da UnB instname:Universidade de Brasília (UnB) instacron:UNB |
instname_str |
Universidade de Brasília (UnB) |
instacron_str |
UNB |
institution |
UNB |
reponame_str |
Repositório Institucional da UnB |
collection |
Repositório Institucional da UnB |
repository.name.fl_str_mv |
Repositório Institucional da UnB - Universidade de Brasília (UnB) |
repository.mail.fl_str_mv |
repositorio@unb.br |
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1814508307038076928 |