Vibration Localization and Anti-Localization of Nonlinear Multi-Support Beams with Support Periodicity Defect

Abstract

A response analysis method for nonlinear beams with spatial distribution parameters and non-periodic supports was developed. The proposed method is implemented in four steps: first, the nonlinear partial differential equation of the beams is transformed into linear partial differential equations with space-varying parameters by using a perturbation method; second, the space-varying parameters are separated into a periodic part and a non-periodic part describing the periodicity defect, and the linear partial differential equations are separated into equations for the periodic and non-periodic parts; third, the equations are converted into ordinary differential equations with multiple modes coupling by using the Galerkin method; fourth, the equations are solved by using a harmonic balance method to obtain vibration responses, which are used to discover dynamic characteristics including the amplitude–frequency relation and spatial mode. The proposed method considers multiple vibration modes in the response analysis of nonlinear non-periodic structures and accounts for mode-coupling effects resulting from structural nonlinearity and parametric non-periodicity. Thus, it can handle nonlinear non-periodic structures with a high parameter-varying wave in wide frequency vibration. In numerical studies, a nonlinear beam with non-periodic supports (resulting in non-periodic distribution parameters or periodicity defect) under harmonic excitations was explored using the proposed method, which revealed some new dynamic response characteristics of this kind of structure and the influences of non-periodic parameters. The characteristics include remarkable variation in frequency response and spatial mode, and in particular, vibration localization and anti-localization. The results have potential applications in vibration control and the support damage detection of nonlinear structures with non-periodic supports.

Response adjustable performance of a visco-elastomer sandwich plate with harmonic parameters and distributed supported masses under random loading

“…In terms of conformation, periodic structures with spatially periodic distribution parameters have special dynamic characteristics, which have been studied on characteristic frequencies, [52][53][54][55][56][57][58][59][60] modal localization and buckling, [61][62][63][64] quasi-periodic distribution parameter effects [65][66][67][68][69][70][71][72][73] and control application [74][75][76][77][78][79][80][81][82][83][84][85][86] based on transfer matrix method, spatial harmonic expansion method, and finite element method, etc. [87][88][89][90][91][92][93] However, the periodicity design has not been applied, but can be applied to improve composite structure dynamics.…”

Response Adjustability Analysis of Partial and Ordinary Differential Coupling System for Visco-Elastomer Sandwich Plate Coupled with Distributed Masses under Random Excitation via Spatial Periodicity Strategy

“…In terms of configuration, periodic structures which have spatially periodic distribution parameters studied exhibit special dynamic characteristics [20,21]. The periodicity design is applicable for improving sandwich structure dynamics.…”