Potential energy of mechanical system dynamics with nonholonomic constraints on the cylinder configuration space

Abstract

The formulation of the dynamics of a mechanical system can be done by the method of the Port Controlled Hamiltonia System (PCHS), but this method still leaves a Lagrange multiplier. Furthermore, the dynamics can be formulated using another method which is more systematic, namely the Routhian Reduction method. The method illustrates a system that is subject to non-holonomic constraints and external force, so that the Lagrange multiplier can be removed from the equation. Before formulating the dynamics of a non-holonomic mechanical system, the researcher will analyze the potential energy that occurs in a system that moves in the cylinder configuration space. Potential energy is the main part that must be completed to formulate the motion system of an object, because Routhian reduction only reviews the kinetic energy and potential energy in a dynamic system. The dynamical system reviewed is an object that moves both translation and rotation with a non-holonomic constraint, namely the Tippe Top (TT). The author analyzes the potential energy of a mechanical system that moves in a cylinder configuration space with non-holonomic constraints. Method in this research is a mathematical theoretical study. This method can reduce the equation TT’s motion with and without friction that moves on the surface of the cylinder clearly in the form of a set of differential equations. According the result of this riset, the potential energy for the TT with non-holonomic constraints that move on the surface in the tube can be determined by U = mg(r(1 − cos ρ ) + (R − acos θ ) cos ρ + asin θ cos φ sin ρ), transforming the TT’s Lagrangian that moves on a flat plane (Cartesian coordinates) to the tube coordinates, with reference to the height of the plane solved by coordinate transformation.