Equations for Nonlinear Analysis of 3D Motions of Beams

The formulation of a set of mathematically consistent differential equations for analyzing nonlinear flexural–flexural–torsional–extensional motions of an Euler–Bernoulli beam is presented. The beam may be mounted on a rotating or on a non–rotating base. A brief discussion on an Euler-like form of the equations is also presented. When the equations are expanded about their equilibrium solution to a desired order in an artificial “bookkeeping parameter” ε, the resulting equations are well suited for a perturbation analysis of the motion. Such analysis discloses a number of important nonlinear phenomena exhibited by the system. Order ε³ equations expanded about the zero equilibrium are also developed here.