Simple but accurate periodic solutions for the nonlinear pendulum equation

Detalhes bibliográficos
Autor(a) principal: Lima, Fábio Menezes de Souza
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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spelling 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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