Free and Forced Wave Motion in a Two-Dimensional Plate with Radial Periodicity

Elisabetta Manconi,Sergey V Sorokin,Matheus Mikael Quartaroli,Rinaldo Garziera

doi:10.3390/app112210948

Elisabetta Manconi, Sergey V Sorokin + Show 2 more

Open Access

https://doi.org/10.3390/app112210948

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Abstract

In many practical engineering situations, a source of vibrations may excite a large and flexible structure such as a ship’s deck, an aeroplane fuselage, a satellite antenna, a wall panel. To avoid transmission of the vibration and structure-borne sound, radial or polar periodicity may be used. In these cases, numerical approaches to study free and forced wave propagation close to the excitation source in polar coordinates are desirable. This is the paper’s aim, where a numerical method based on Floquet-theory and the FE discretision of a finite slice of the radial periodic structure is presented and verified. Only a small slice of the structure is analysed, which is approximated using piecewise Cartesian segments. Wave characteristics in each segment are obtained by the theory of wave propagation in periodic Cartesian structures and Finite Element analysis, while wave amplitude change due to the changes in the geometry of the slice is accommodated in the model assuming that the energy flow through the segments is the same. Forced response of the structure is then evaluated in the wave domain. Results are verified for an infinite isotropic thin plate excited by a point harmonic force. A plate with a periodic radial change of thickness is then studied. Free waves propagation are shown, and the forced response in the nearfield is evaluated, showing the validity of the method and the computational advantage compared to FE harmonic analysis for infinite structures.

Highlights

Starting from the milestone book written by Leon Brillouin [1], wave propagation in periodic media has been a subject extensively studied
Amongst the numerical methods that could be used to investigate wave propagation problems, the Wave Finite Element (WFE) technique has several desirable features: it can be applied both to continuous and periodic structures; it exploits Bloch-Floquet theory and the versatility of standard FE analysis of a very small part of the structure; it allows the study of waveguides
Following a previous study by the same authors [33], this paper presents a simplified adaptation of this WFE technique to structures in polar coordinates exhibiting periodicity in the radial directions

Summary

Introduction

Starting from the milestone book written by Leon Brillouin [1], wave propagation in periodic media has been a subject extensively studied. In many practical engineering situations, a source of vibrations may excite a large and flexible structure such as a ship’s deck, an aeroplane fuselage, a satellite antenna, a wall panel In these cases, radial periodicity (e.g., as a sequence of annuli with alternating properties) may be used to reduce transmission of the vibration and structure-borne sound, and numerical approaches to study free and forced wave propagation in polar coordinates are desirable. Amongst the numerical methods that could be used to investigate wave propagation problems, the WFE technique has several desirable features: it can be applied both to continuous and periodic structures; it exploits Bloch-Floquet theory and the versatility of standard FE analysis of a very small part of the structure; it allows the study of waveguides.

Stepwise WFE Approximation of a Radially Periodic Plate

Free Wave Propagation Characteristics

Forced Wave Amplitude

Coupling of the Segments and Wave Amplitude Decay

Numerical Examples