Simple Linearized Equation of Motion for Spinning Tops on a Table

Abstract

This paper deals with deriving a simple linearized equation of motion for axisymmetric tops spinning on a table; the equation includes the effects of both friction and gravitation. The derived equation is a single second-order equation with complex number coefficients. The main variable is the inclination angle of the spinning axis, which is expressed by a single complex number. No Euler's angle is used. The precondition for linearization is that both the angular momentum vector and the spinning axis are close to the zenith direction. The friction is assumed to be viscous, and the friction torque can be expressed by using the height of the gravity center and the radius of the curvature at the contact point to the table. Two eigenvalues for the equation give a deep insight into the attitude behavior. The root-locus method was applied to explain the spinning motion stability of various axisymmetric bodies. An analytical solution for a horizontally suspended spinning top is given to show that the locus of the spinning axis is a trochoid.