Formulation of Kirchhoff Rod Based on Quasi-coordinates

Abstract

The quasi-coordinates are applied to formulate Kirchhoff's rod. The potential energy of the rod expressed by the quasi-coordinates has a similar form as the kinetic energy and complementary kinetic energy in dynamics. The conjugate quasi-momentum is defined and the canonical equations due to the quasi-coordinates are given. Kirchhoff's equations can be derived directly from Boltzman-Hamel's equations or its canonical form with arc length s as independent variables. Lagrange's theorem is extended to determine the stability of equilibrium con-figuration of the elastic rod, and is proved using the Lyapunov's direct method. It is noticed that the condition of stability has a different physical explanation than in dynamics.