Nonlinear dynamics of a flexible beam in a central gravitational field—I. Equations of motion

Abstract

The complete nonlinear differential equations governing the nonlinear motions of a beam able to undergo bending and pitching in space, are formulated in this paper. The formulation is based on a variational principle and accounts for all the nonlinearities due to deformation and gravity gradient effects. The nonlinearities due to deformation arise due to geometric effects, which consist of nonlinear curvature and nonlinear inertia terms. Expanded equations governing the nonlinear perturbed motion about an equilibrium are also developed for the case when the beam is in circular orbit. Such equations are suited for a perturbation analysis of the motion, and nonlinearities up to cubic order in a bookkeeping parameter are retained in them. Nonlinear motions involving interactions between bending and pitching of the beam are investigated in Part II of this work using the equations developed here.