PEMODELAN DINAMIKA TIPPE TOP (TT) DENGAN KENDALA NON-HOLONOMIK BERBASIS KOMPUTASI FISIKA PADA BIDANG DATAR (R^2×SO(3))

Pemodelan sistem dinamik dalam real time sangat penting dalam kemajuan teknologi otomatis yang berkembang pesat saat ini, seperti metode perencanaan sistem robotic. Artikel ini menjelaskan sistem dinamik benda tegar dengan kendala non-holonomic pada ruang konfigurasi . Metode yang digunakan adalah Motion Planning Network dan simulasi numeric dengan komputasi fisika yang dapat digunakan untuk sistem benda non-holonomik yang bergerak secara real-time dengan Pendekatan Jellet Invarian (JI). Pendekatan JI dapat menghasilkan persamaan sistem gerak dan mengevaluasi simulasi model benda dengan kendala non holonomik dan juga menampilkan hasil eksperimen dinamika benda tegar dalam ruang konfigurasi . Sistem gerak benda dengan kendala non holonomik yang digunakan adalah Tippe top (TT). TT adalah mainan yang mirip seperti gasing yang jika diputar dapat membalik sendiri dengan batangnya. Penulis berhasil mendeskripsikan dinamika gerak TT secara real time dengan syarat awal bervariasi pada ruang konfigurasi .

The Mathematical Model for the Tippe Top Inversion

The dynamics of rotating objects is an area of classical mechanics that has many unsolved problems. Among these problems are the gyroscopic effects manifested by the spinning objects of different forms. One of them is the Tippe top designed as the truncated sphere which is fitted with a short, cylindrical rod for rotation. The unexplainable gyroscopic effect of the Tippe top is manifested by its inversion towards the support surface. Researchers tried to describe this gyroscopic effect for two centuries, but all modelings were on the level of assumptions. It is natural because the Tippe top has a more complex design than the simple spinning disc, which gyroscopic effects did not have an analytical solution until recent time. The latest research, in the area of gyroscopic effects, reveals the action of the system of several interrelated inertial torques on any spinning object. The gyroscopic inertial torques are generated by their rotating mass. These inertial torques and the variable ratio of the angular velocities of the spinning object around axes of rotations constitute the fundamental principles of gyroscope theory. These physical principles of dynamics of rotating objects enable to description and compute of any gyroscopic effects and also the Tippe top inversion.

Utilization of Maple-based Physics Computation in Determining the Dynamics of Tippe Top

Tippe top is an example of simple moving system of rigid body with non-holonomic constraint, but the analysis of this system is not simple. A tippe top equation has been derived with Routhian reduction method and Poincaré equation, and physics computation in finding numeric solution of the dynamics of the tippe top has also been utilized by using Maple program. However, the Poincaré equation required that quasi-coordinate of the quasi-velocity is found, while in the case of the dynamics of tippe top, there is not any exact solution of the quasi-coordinate of the quasi-velocity was found. Therefore, the tippe top equation should be reduced to solve the problem. In this research, Routhian reduction was employed so that the Routhian reduction-based Poincaré equation was used to derive the tippe top equation. The method was able to derive a tippe top equation on a flat plane and tube inner surface clearly represented differential equations.