THE FIGURE-OF-8 LIBRATIONS OF THE GRAVITY GRADIENT PENDULUM AND MODES OF AN ORBITING TETHER
An algorithm is presented for the Hill-Poincaré analytical continuation of the out-of-plane normal mode of the gravity gradient pendulum. The Poincaré-Lindstedt solution employs 17 Poisson series and 24 recursion relations and was evaluated to the 50th order on a CRAY. The trajectories of the nonlin...
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| Veröffentlicht in: | Quarterly of applied mathematics 1988-12, Vol.46 (4), p.637-663 |
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| container_title | Quarterly of applied mathematics |
| container_volume | 46 |
| creator | MELVIN, PETER J. |
| description | An algorithm is presented for the Hill-Poincaré analytical continuation of the out-of-plane normal mode of the gravity gradient pendulum. The Poincaré-Lindstedt solution employs 17 Poisson series and 24 recursion relations and was evaluated to the 50th order on a CRAY. The trajectories of the nonlinear normal modes are figures-of-8 on the unit sphere which can be computed nearly to the orbit normal. Numerical integrations indicate further that 1) initial conditions computed at the nadir can be used to generate figures-of-8 over the pole, 2) the single hemispherical figures-of-8 appear to be stable at large amplitudes, and 3) the gravity gradient pendulum has chaotic solutions. A theory is developed for the linear normal modes of a tethered satellite, and the eigenvalues are found for the rosary tether. |
| doi_str_mv | 10.1090/qam/973381 |
| format | Article |
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The Poincaré-Lindstedt solution employs 17 Poisson series and 24 recursion relations and was evaluated to the 50th order on a CRAY. The trajectories of the nonlinear normal modes are figures-of-8 on the unit sphere which can be computed nearly to the orbit normal. Numerical integrations indicate further that 1) initial conditions computed at the nadir can be used to generate figures-of-8 over the pole, 2) the single hemispherical figures-of-8 appear to be stable at large amplitudes, and 3) the gravity gradient pendulum has chaotic solutions. 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The Poincaré-Lindstedt solution employs 17 Poisson series and 24 recursion relations and was evaluated to the 50th order on a CRAY. The trajectories of the nonlinear normal modes are figures-of-8 on the unit sphere which can be computed nearly to the orbit normal. Numerical integrations indicate further that 1) initial conditions computed at the nadir can be used to generate figures-of-8 over the pole, 2) the single hemispherical figures-of-8 appear to be stable at large amplitudes, and 3) the gravity gradient pendulum has chaotic solutions. A theory is developed for the linear normal modes of a tethered satellite, and the eigenvalues are found for the rosary tether.</description><subject>Algorithms</subject><subject>Center of mass</subject><subject>Coefficients</subject><subject>Eigenvalues</subject><subject>Equations of motion</subject><subject>Gravity</subject><subject>Kinetics</subject><subject>Libration</subject><subject>Pendulums</subject><subject>Trajectories</subject><issn>0033-569X</issn><issn>1552-4485</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>1988</creationdate><recordtype>article</recordtype><recordid>eNo9kM9LwzAYhoMoOKcX70LOQlzS_GqO3Zp2ga6VmorioWRdAw7HtN3F_97WiqeXj_d5v8MDwC3BDwQrvPhyh4WSlIbkDMwI5wFiLOTnYIYxpYgL9XIJrvp-P5xDi2fgza41TExalRoVCQphZpZlZE2RP8EigWObltGzsa9jxkbnFj7qPK6yagOjPIabIta_aJTDolwaa_IUWj0My2tw4d1H39785RxUibarNcqK1KyiDDVBKE-oFW7rmQh802JHcNgoLpuACe-F9B6rxsktF0Qq73eUO7djJPBEcceIbOmW0jm4n_423bHvu9bXn937wXXfNcH1qKUetNSTlgG-m-B9fzp2_ySjgkoeUPoDqQpXTw</recordid><startdate>19881201</startdate><enddate>19881201</enddate><creator>MELVIN, PETER J.</creator><general>Brown University</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>19881201</creationdate><title>THE FIGURE-OF-8 LIBRATIONS OF THE GRAVITY GRADIENT PENDULUM AND MODES OF AN ORBITING TETHER</title><author>MELVIN, PETER J.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c287t-e6abf462fce0a108c957c246ff67ff09ca7b56179ffd35aad412f195a417e3b33</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>1988</creationdate><topic>Algorithms</topic><topic>Center of mass</topic><topic>Coefficients</topic><topic>Eigenvalues</topic><topic>Equations of motion</topic><topic>Gravity</topic><topic>Kinetics</topic><topic>Libration</topic><topic>Pendulums</topic><topic>Trajectories</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>MELVIN, PETER J.</creatorcontrib><collection>CrossRef</collection><jtitle>Quarterly of applied mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>MELVIN, PETER J.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>THE FIGURE-OF-8 LIBRATIONS OF THE GRAVITY GRADIENT PENDULUM AND MODES OF AN ORBITING TETHER</atitle><jtitle>Quarterly of applied mathematics</jtitle><date>1988-12-01</date><risdate>1988</risdate><volume>46</volume><issue>4</issue><spage>637</spage><epage>663</epage><pages>637-663</pages><issn>0033-569X</issn><eissn>1552-4485</eissn><abstract>An algorithm is presented for the Hill-Poincaré analytical continuation of the out-of-plane normal mode of the gravity gradient pendulum. The Poincaré-Lindstedt solution employs 17 Poisson series and 24 recursion relations and was evaluated to the 50th order on a CRAY. The trajectories of the nonlinear normal modes are figures-of-8 on the unit sphere which can be computed nearly to the orbit normal. Numerical integrations indicate further that 1) initial conditions computed at the nadir can be used to generate figures-of-8 over the pole, 2) the single hemispherical figures-of-8 appear to be stable at large amplitudes, and 3) the gravity gradient pendulum has chaotic solutions. A theory is developed for the linear normal modes of a tethered satellite, and the eigenvalues are found for the rosary tether.</abstract><pub>Brown University</pub><doi>10.1090/qam/973381</doi><tpages>27</tpages><oa>free_for_read</oa></addata></record> |
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| identifier | ISSN: 0033-569X |
| ispartof | Quarterly of applied mathematics, 1988-12, Vol.46 (4), p.637-663 |
| issn | 0033-569X 1552-4485 |
| language | eng |
| recordid | cdi_crossref_primary_10_1090_qam_973381 |
| source | American Mathematical Society Publications (Freely Accessible); JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing; American Mathematical Society Publications; EZB-FREE-00999 freely available EZB journals |
| subjects | Algorithms Center of mass Coefficients Eigenvalues Equations of motion Gravity Kinetics Libration Pendulums Trajectories |
| title | THE FIGURE-OF-8 LIBRATIONS OF THE GRAVITY GRADIENT PENDULUM AND MODES OF AN ORBITING TETHER |
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