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.
Highlights
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Many engineering structures such as continuous girder viaducts and synchrotron radiation bridges can be simplified as a beam with multi-supports in dynamics analysis
Numerous studies have reported characteristic frequencies [1,2,3,4], modal localization and buckling [5,6,7,8,9,10,11,12,13,14], quasi-periodic distribution parameter effects [15,16,17,18] and application [19,20,21,22,23,24,25,26,27] based on transfer matrix method, spatial (Bloch) harmonic expansion method, (Floquet–Bloch and Galerkin) double expansion method and finite element method, etc
The ordinary differential equations are solved by using a harmonic balance method to obtain vibration responses of the beam with non-periodic parameters under harmonic excitations, which are used for characteristics analysis of amplitude–frequency relation and spatial mode
Summary
Many engineering structures such as continuous girder viaducts and synchrotron radiation bridges can be simplified as a beam with multi-supports in dynamics analysis. The ordinary differential equations are solved by using a harmonic balance method to obtain vibration responses of the beam with non-periodic parameters under harmonic excitations, which are used for characteristics analysis of amplitude–frequency relation and spatial mode.
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