Dynamics of a mass–spring–pendulum system with vastly different frequencies
We investigate the dynamics of a simple pendulum coupled to a horizontal mass–spring system. The spring is assumed to have a very large stiffness value such that the natural frequency of the mass–spring oscillator, when uncoupled from the pendulum, is an order of magnitude larger than that of the os...
Saved in:
| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | article |
| Published: |
2012
|
| Online Access: | http://hdl.handle.net/10725/6289 http://dx.doi.org/10.1007/s11071-012-0428-9 http://libraries.lau.edu.lb/research/laur/terms-of-use/articles.php https://link.springer.com/content/pdf/10.1007%2Fs11071-012-0428-9.pdf |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1864513478966902784 |
|---|---|
| author | Sheheitli, Hiba |
| author2 | Rand, Richard H. |
| author2_role | author |
| author_facet | Sheheitli, Hiba Rand, Richard H. |
| author_role | author |
| dc.creator.none.fl_str_mv | Sheheitli, Hiba Rand, Richard H. |
| dc.date.none.fl_str_mv | 2012 2017-09-28T12:50:03Z 2017-09-28T12:50:03Z 2017-09-28 |
| dc.identifier.none.fl_str_mv | 1573-269X http://hdl.handle.net/10725/6289 http://dx.doi.org/10.1007/s11071-012-0428-9 Sheheitli, H., & Rand, R. H. (2012). Dynamics of a mass–spring–pendulum system with vastly different frequencies. Nonlinear Dynamics, 70(1), 25-41. http://libraries.lau.edu.lb/research/laur/terms-of-use/articles.php https://link.springer.com/content/pdf/10.1007%2Fs11071-012-0428-9.pdf |
| dc.language.none.fl_str_mv | en |
| dc.relation.none.fl_str_mv | Nonlinear Dynamics |
| dc.rights.*.fl_str_mv | info:eu-repo/semantics/openAccess |
| dc.title.none.fl_str_mv | Dynamics of a mass–spring–pendulum system with vastly different frequencies |
| dc.type.none.fl_str_mv | Article info:eu-repo/semantics/publishedVersion info:eu-repo/semantics/article |
| description | We investigate the dynamics of a simple pendulum coupled to a horizontal mass–spring system. The spring is assumed to have a very large stiffness value such that the natural frequency of the mass–spring oscillator, when uncoupled from the pendulum, is an order of magnitude larger than that of the oscillations of the pendulum. The leading order dynamics of the autonomous coupled system is studied using the method of Direct Partition of Motion (DPM), in conjunction with a rescaling of fast time in a manner that is inspired by the WKB method. We particularly study the motions in which the amplitude of the motion of the harmonic oscillator is an order of magnitude smaller than that of the pendulum. In this regime, a pitchfork bifurcation of periodic orbits is found to occur for energy values larger that a critical value. The bifurcation gives rise to nonlocal periodic and quasi-periodic orbits in which the pendulum oscillates about an angle between zero and π/2 from the down right position. The bifurcating periodic orbits are nonlinear normal modes of the coupled system and correspond to fixed points of a Poincare map. An approximate expression for the value of the new fixed points of the map is obtained. These formal analytic results are confirmed by comparison with numerical integration. |
| eu_rights_str_mv | openAccess |
| format | article |
| id | LAURepo_e17c6d0fca86b8c76b2a3a08ac04a418 |
| identifier_str_mv | 1573-269X Sheheitli, H., & Rand, R. H. (2012). Dynamics of a mass–spring–pendulum system with vastly different frequencies. Nonlinear Dynamics, 70(1), 25-41. |
| language_invalid_str_mv | en |
| network_acronym_str | LAURepo |
| network_name_str | Lebanese American University repository |
| oai_identifier_str | oai:laur.lau.edu.lb:10725/6289 |
| publishDate | 2012 |
| repository.mail.fl_str_mv | |
| repository.name.fl_str_mv | |
| repository_id_str | |
| spelling | Dynamics of a mass–spring–pendulum system with vastly different frequenciesSheheitli, HibaRand, Richard H.We investigate the dynamics of a simple pendulum coupled to a horizontal mass–spring system. The spring is assumed to have a very large stiffness value such that the natural frequency of the mass–spring oscillator, when uncoupled from the pendulum, is an order of magnitude larger than that of the oscillations of the pendulum. The leading order dynamics of the autonomous coupled system is studied using the method of Direct Partition of Motion (DPM), in conjunction with a rescaling of fast time in a manner that is inspired by the WKB method. We particularly study the motions in which the amplitude of the motion of the harmonic oscillator is an order of magnitude smaller than that of the pendulum. In this regime, a pitchfork bifurcation of periodic orbits is found to occur for energy values larger that a critical value. The bifurcation gives rise to nonlocal periodic and quasi-periodic orbits in which the pendulum oscillates about an angle between zero and π/2 from the down right position. The bifurcating periodic orbits are nonlinear normal modes of the coupled system and correspond to fixed points of a Poincare map. An approximate expression for the value of the new fixed points of the map is obtained. These formal analytic results are confirmed by comparison with numerical integration.PublishedN/A2017-09-28T12:50:03Z2017-09-28T12:50:03Z20122017-09-28Articleinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article1573-269Xhttp://hdl.handle.net/10725/6289http://dx.doi.org/10.1007/s11071-012-0428-9Sheheitli, H., & Rand, R. H. (2012). Dynamics of a mass–spring–pendulum system with vastly different frequencies. Nonlinear Dynamics, 70(1), 25-41.http://libraries.lau.edu.lb/research/laur/terms-of-use/articles.phphttps://link.springer.com/content/pdf/10.1007%2Fs11071-012-0428-9.pdfenNonlinear Dynamicsinfo:eu-repo/semantics/openAccessoai:laur.lau.edu.lb:10725/62892021-03-19T10:03:25Z |
| spellingShingle | Dynamics of a mass–spring–pendulum system with vastly different frequencies Sheheitli, Hiba |
| status_str | publishedVersion |
| title | Dynamics of a mass–spring–pendulum system with vastly different frequencies |
| title_full | Dynamics of a mass–spring–pendulum system with vastly different frequencies |
| title_fullStr | Dynamics of a mass–spring–pendulum system with vastly different frequencies |
| title_full_unstemmed | Dynamics of a mass–spring–pendulum system with vastly different frequencies |
| title_short | Dynamics of a mass–spring–pendulum system with vastly different frequencies |
| title_sort | Dynamics of a mass–spring–pendulum system with vastly different frequencies |
| url | http://hdl.handle.net/10725/6289 http://dx.doi.org/10.1007/s11071-012-0428-9 http://libraries.lau.edu.lb/research/laur/terms-of-use/articles.php https://link.springer.com/content/pdf/10.1007%2Fs11071-012-0428-9.pdf |