A reduced-order analytical model for the nonlinear dynamics of a class of flexible multi-beam structures

Abstract

Nonlinearities in the differential equations of motion of dynamical systems can play, under certain conditions, such a dominant role that the motion described by linearized differential equations bears no resemblance to the actual motion exhibited by the systems. For a structure, nonlinearities are due to material behavior and to deformation. The latter are called “geometric nonlinearities” and, even for linear (i.e., Hookean) materials and small deformations, their effect can be dramatic. To investigate the nonlinear behavior of a dynamical system by making use of analytical techniques, one must start the analysis by formulating a set of mathematically consistent differential equations of motion for the system. Furthermore, the equations must be cast in a form that makes them amenable to the application of known analytical methods, such as perturbation techniques, to investigate the motion. The work presented in this paper addresses the formulation of such equations for a class of multi-beam structures. Each beam in the structure may have arbitrary cross section variation along its span, but behaves as inextensional. The structure may have any number of beams and supports, and may carry any number of lumped masses along its span.