Simple but accurate periodic solutions for the nonlinear pendulum equation
محفوظ في:
| المؤلف الرئيسي: | |
|---|---|
| تاريخ النشر: | 2019 |
| التنسيق: | Article |
| اللغة: | eng |
| المصدر: | Repositório Institucional da UnB |
| Download full: | https://repositorio.unb.br/handle/10482/36547 https://doi.org/10.1590/1806-9126-rbef-2018-0202 http://orcid.org/0000-0001-5884-6621 |
الملخص: | 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. |
| _version_ | 1871442449923047424 |
|---|---|
| author | Lima, Fábio Menezes de Souza |
| author_browse | Lima, Fábio Menezes de Souza |
| author_facet | Lima, Fábio Menezes de Souza |
| author_role | author |
| bitstream.checksum.fl_str_mv | 42d81ec6a7036365f9ed6f7845c01034 |
| bitstream.checksumAlgorithm.fl_str_mv | MD5 |
| bitstream.url.fl_str_mv | http://repositorio2.unb.br/jspui/bitstream/10482/36547/1/ARTIGO_SimpleAccuratePeriodic.pdf |
| collection | Repositório Institucional da UnB |
| dc.contributor.author.fl_str_mv | Lima, Fábio Menezes de Souza |
| dc.date.accessioned.fl_str_mv | 2020-01-24T10:31:44Z |
| dc.date.available.fl_str_mv | 2020-01-24T10:31:44Z |
| dc.date.issued.fl_str_mv | 2019 |
| dc.identifier.citation.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. |
| dc.identifier.doi.pt_BR.fl_str_mv | https://doi.org/10.1590/1806-9126-rbef-2018-0202 |
| dc.identifier.orcid.none.fl_str_mv | http://orcid.org/0000-0001-5884-6621 |
| dc.identifier.uri.fl_str_mv | https://repositorio.unb.br/handle/10482/36547 |
| dc.language.iso.fl_str_mv | eng |
| dc.publisher.none.fl_str_mv | Sociedade Brasileira de Física |
| dc.rights.driver.fl_str_mv | (CC BY) - LIcença Creative Commons. info:eu-repo/semantics/openAccess |
| dc.source.none.fl_str_mv | reponame:Repositório Institucional da UnB instname:Universidade de Brasília (UnB) instacron:UNB |
| dc.subject.keyword.pt_BR.fl_str_mv | Pêndulo Vibração Fourier, Séries de Funções elípticas |
| dc.title.pt_BR.fl_str_mv | Simple but accurate periodic solutions for the nonlinear pendulum equation |
| dc.type.driver.fl_str_mv | info:eu-repo/semantics/article |
| dc.type.status.fl_str_mv | info:eu-repo/semantics/publishedVersion |
| 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. |
| eu_rights_str_mv | openAccess |
| format | article |
| id | UNB_ab70d49cc2c8fd1a5db75fa96a7980da |
| 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. |
| instacron_str | UNB |
| institution | UNB |
| instname_str | Universidade de Brasília (UnB) |
| language | eng |
| network_acronym_str | UNB |
| network_name_str | Repositório Institucional da UnB |
| oai_identifier_str | oai:repositorio.unb.br:10482/36547 |
| publishDate | 2019 |
| publishDateSort | 2019 |
| publisher.none.fl_str_mv | Sociedade Brasileira de Física |
| reponame_str | Repositório Institucional da UnB |
| repository.mail.fl_str_mv | repositorio@unb.br |
| repository.name.fl_str_mv | Repositório Institucional da UnB - Universidade de Brasília (UnB) |
| repository_id_str | |
| rights_invalid_str_mv | (CC BY) - LIcença Creative Commons. |
| spelling | Lima, Fábio Menezes de Souza2020-01-24T10:31:44Z2020-01-24T10:31:44Z2019Lima, 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/0000-0001-5884-6621Sociedade Brasileira de Física(CC BY) - LIcença Creative Commons.info:eu-repo/semantics/openAccessSimple but accurate periodic solutions for the nonlinear pendulum equationinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articlePênduloVibraçãoFourier, 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.engreponame:Repositório Institucional da UnBinstname:Universidade de Brasília (UnB)instacron:UNBORIGINALARTIGO_SimpleAccuratePeriodic.pdfapplication/pdf4015392http://repositorio2.unb.br/jspui/bitstream/10482/36547/1/ARTIGO_SimpleAccuratePeriodic.pdf42d81ec6a7036365f9ed6f7845c01034MD51open access10482/365472023-05-26 21:19:55.691open accessoai: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) |
| spellingShingle | Simple but accurate periodic solutions for the nonlinear pendulum equation Lima, Fábio Menezes de Souza Pêndulo Vibração Fourier, Séries de Funções elípticas |
| status_str | publishedVersion |
| title | 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_short | Simple but accurate periodic solutions for the nonlinear pendulum equation |
| title_sort | Simple but accurate periodic solutions for the nonlinear pendulum equation |
| topic | Pêndulo Vibração Fourier, Séries de Funções elípticas |
| 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 |