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Symmetry theory-based approaches for the computation of mode shapes of vibrating structures

Grant number: 21/12894-2
Support type:Scholarships in Brazil - Doctorate (Direct)
Effective date (Start): March 01, 2022
Effective date (End): February 28, 2025
Field of knowledge:Engineering - Mechanical Engineering - Mechanics of Solids
Principal researcher:Samuel da Silva
Grantee:Afonso Willian Nunes
Home Institution: Faculdade de Engenharia (FEIS). Universidade Estadual Paulista (UNESP). Campus de Ilha Solteira. Ilha Solteira , SP, Brazil

Abstract

The study of vibrating elastic bodies plays a crucial role in the dynamic analysis of Engineering problems. For that, displacement patterns of vibrating systems, i.e., mode shapes, have shown great importance for assessing their motion behavior, whose modeling arises from differential equations. Symmetry methods assist the exact resolution of differential equations through analytical transformations such as order reductions, domain compression, and linearizations. This research proposal addresses symmetry methods for computing original mode shapes of uniform and non-uniform rods, beams, and plates, aiming to expand the formulation for periodic structures and metamaterial applications. Prior investigation shows that the Lie symmetry method accomplishes new exact mode shapes of non-uniform rods and accurate ones for uniform Euler-Bernoulli beams, free from round-off errors when numerically assessed at all frequencies. Also, this method provides expressions for mode shapes of uniform rectangular plates, namely invariant solutions, that may be superimposed to develop original mode shape solutions. On the other hand, the Lie symmetry method has been found inefficient for other generic non-uniform cases due to their modeling equations with arbitrary functions. This proposal suggests an extension of the previous mode shape studies by using better-conditioned symmetry methods for differential equations with arbitrary functions, e.g., the generalized symmetry method. Within this framework, new vibroacoustic applications may be achieved by this research, enhancing prediction, simulation, and design capabilities of real complex problems. (AU)

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