Vibration analysis of beam structures with localized nonlinearities by a wave approach

Abstract

This paper presents a general wave propagation approach for the vibration analysis of structures consisting of beam elements with localized nonlinearities. The scattering of a wave at a nonlinear discontinuity creates multi-harmonic waves. This forms the basis of the model formulation in which the displacement in each element composing the structure is defined as a sum of waves associated with the fundamental frequency and its harmonics. The coupling between separate elements is expressed using transmission coefficients function of the harmonic number and includes propagating and decaying near field waves whose amplitudes are the unknowns of the problem. With the use of harmonic balance, a characteristic algebraic equation of motion is derived. The general systematic methodology is demonstrated and detailed on beam-like systems with nonlinear boundaries for validation against a conventional finite element analysis. To illustrate the effectiveness of the approach on a more practical case, the method is also applied to a simplified model of a turbomachinery blade with underplatform friction dampers. The predictions of stick-slip effect and global nonlinear damping behavior made using the proposed approach are shown to be in excellent agreement with the results of numerical time integration with the advantage of reduced computational costs.