Approximation of Infinitesimal Rotations in Calculus of Variations

Abstract

This paper focuses on problems dealing with very small angular displacements, i.e., infinitesimal or micro rotations, where these rotations are desired to be approximately treated as Euclidean vectors. For such problems, an appropriate and consistent approach to approximate the rotation matrix, the angular velocity and acceleration vectors, and the virtual rotation vector, up to the first order, is presented and the calculus of variations (or Hamilton’s principle) when applied to such problems is characterized. Also, a discrepancy observed in previous works dealing with the infinitesimal rotations, where second- or higher-order approximations have been employed, is reviewed. The consistent and comprehensive approach for the first-order approximation of infinitesimal rotations, presented in this paper, is employed to derive the dynamic equations of a symmetric spinning top and a gyroelastic continuum and is proved to result in the correct and complete dynamic equations.