DINAMIKA SISTEM MEKANIK TAK HOLONOMIK DENGAN RUANG KONFIGURASI S2 x T3 (SNAKEBOARD)

MUHARANI ASNAL, Dr. rer.nat. M.F. Rosyid

2011 | Tesis | S2 Ilmu Fisika

Secara matematis, suatu kendala sistem mekanik pada keragaman Riemannan diwakili oleh himpunan forma-1 yang tidak merosot. Kendala kemudian menginduksi suatu distribusi medan vektor yang melenyapkan semua nilai kendala forma-1. Ketika distribusi tersebut tidak involutive (atau dapat diintegralkan), maka kendala tersebut disebut tak holonomik. Gerak snakeboard pada arena melengkung adalah contoh sistem mekanik dengan kendala tak holonomik. Gerak snakeboard pada permukaan bola bagian dalam dipelajari dengan asumsi jari-jari arena permukaan bola sangat besar dan energi yang terlibat cukup kecil sehingga gerak snakeboard terbatas di sekitar titik nadhir permukaan bola. Forma-forma kendala sistem diturunkan melalui metode geometris. Persamaan dinamika sistem awalnya diturunkan menggunakan Port-Controlled Hamiltonian System dan metode koneksi Levi-Civita terkendala pada keragaman konfigurasi. Metode PCHS mengalami kesulitan dalam penentuan basis yang melenyapkan forma-forma kendala dan mendiagonalkan metrik inersianya. Sebaliknya, metode koneksi Levi-Civita terkendala memberikan hasil yang lebih sistematis pada persamaan gerak snakeboard pada permukaan bola bagian dalam.

Mathematically, a constraint in a mechanical system living on a Riemannian manifold are represented by a set of 1-forms which are non-degenerate. The constraint then induces a distribution of vector fields in which the values of all constraint 1-forms vanish. When the associated distribution is not involutive (or integrabel), then the constraint is said to be nonholonomic. The motion of a snakeboard on a (curved) arena is an instance of a mechanical system with a nonholonomic constraint. We have studied the motion of snakeboard on the internal surface of a sphere in which we assumed that the radius of the spherical arena is so great and the involved energy is small enough so that the motion of the snakeboard is limited around the nadir point of the sphere. We have derived the constraint 1-forms of the system. The dynamical equations of the system were then derived by making use of PCHS the so-called Port-Controlled Hamiltonian System as well as constrained Leci-Civita connection method on the configuration manifold. The PCHS method faced some difficulties concerning the determination of basis vanishing the constraint one-forms and diagonalizing the inertia metric. On the other hand, the constrained Levi-Civita connection method worked systematically resulting in the equation of motion of the snake board on the spherical arena.

Kata Kunci : keragaman, kendala, dinamika, snakeboard.