Coupling Finite Element Method with Incremental Harmonic Balance Method for Analysis of Nonlinear Vibrations of Stepped Beams and Frames

Abstract

The developed numerical methodology allows analysis of nonlinear vibrations of stepped beams and frames with contrast elastic properties. The principle of virtual work is applied to derive nonlinear ordinary matrix differential equations of longitudinal and transverse vibrations in terms of nodal displacement vectors. The steady-state periodic solution is computed by using the IHB method with time scale in the form of truncated Fourier series. The IHB method consists of the Newton-Raphson iterative procedure in which the linearized matrix partial differential equations are derived and the Galerkin procedure, where the corresponding variational equations are integrated and resulting linear algebraic equations are solved for unknown vectors of increments of Fourier coefficients. By exchanging the Newton-Raphson iterative procedure with the augmentation process in which some system parameter is changed, the various branches of solutions can be traced for fundamental, superharmonic and subharmonic resonances of beams.