General Reduced Order Analytical Model for Nonlinear Dynamic Analyses of Beams With or Without Lumped Masses

A set of reduced order differential equations of motion that are suited for analyzing the nonlinear dynamics of beams subjected to external excitations is developed using a variational formulation. The beam may have arbitrary property variations along its span, may carry any number of concentrated masses, and may have multiple supports. It may also be subjected to a base excitation in the form of a prescribed displacement imposed to the supports. The distributed and/or concentrated forces acting on the system may have a nonzero time average so that the equilibrium solution of the system does not necessarily coincide with its undeformed state. Because the first approximation to the elastic deformation of the beam is governed, in general, by partial differential equations with variable coefficients, the solution for the bending displacements at that level is obtained numerically. An analytical methodology is used to formulate, in a mathematically consistent manner, the reduced order nonlinear differential equations explicitly. Specific examples are then used in order to assess the combined effect of the nonlinear terms on the dynamic response of a beam subjected to both static and dynamic loads.