A Motion of Top by Numerical Calculation(General)

概要

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We describe the results of numerical calculation for the motion of a top, under taking acount of the sliding friction force between the edge of the top and the floor. New angular velocities α and β are introduced for generating an orthogonal matrix enabling the transformation between principal angular velocities and a set of α, θ and β. Time derivative of Euler angle φ is represented by a linear combination of these new angular velocities. This transformation and linear combination allows us to remove the singularities and spurious behavior near Euler angle θ = 0 or θ = π and to produce the trajectory of three angular velocities continuously. The orthogonal transformation is successfully introduced to give no difficulty of divergence. A new method is proposed which should lead to correct equations of motion even for a rigid body and provide improved numerical stability and precision for computer simulation studies. Application of this method to the motion of a top enables us to describe time variation of angular velocities, stability of solution for motion and their continuity in real time and space. We also rediscovered the existence of a constant of motion, the so-called Jellett constant, which is independent of the existence of friction and can be written in different forms.
社団法人日本物理学会の論文
2004-08-15