2020/03/20 by Matthieu Gallezot, Gallezot, Matthieu, Fabien Treyssède +3
Engineering · Physics and Astronomy · #Advanced Fiber Optic Sensors #Applied Physics (physics.app-ph) #FOS: Physical sciences #Fluid Dynamics and Vibration Analysis #Mechanical and Optical Resonators #Railway Engineering and Dynamics #Soft Condensed Matter (cond-mat.soft) #Vibration and Dynamic Analysis
paper · pdf · doi:10.48550/arxiv.2004.11887
openalex publication_date 2020/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A numerical model is proposed to compute the eigenmodes and the forced\nresponse of multilayered elastic spheres. The main idea is to describe\nanalytically the problem along the angular coordinates with spherical harmonics\nand to discretize the radial direction with one-dimensional finite elements.\nThe proper test function must be carefully chosen so that both vector and\ntensor spherical harmonics orthogonality relationships can be used. The\nproposed approach yields a general one-dimensional formulation with a fully\nanalytical description of the angular behaviour, suitable for any interpolating\ntechnique. A linear eigenvalue problem, simple and fast to solve, is then\nobtained. The eigensolutions are the spheroidal and torsional modes. They are\nfavourably compared with literature results for a homogeneous sphere. The\neigensolutions are superposed to compute explicitly the forced response. The\nlatter is used to reconstruct the propagation of surfaces waves. In particular,\nthe collimation of a Rayleigh wave (non-diffracted surface wave propagating\nwith a quasi-constant width) excited by a line source in a homogeneous sphere\nis recovered with the model. Based on the vibration eigenmodes, a modal\nanalysis shows that such a wave is a superposition of fundamental spheroidal\nmodes with a displacement confined at the equator of the sphere. These modes\nare the so-called Rayleigh modes, of sectoral type and high polar wavenumbers.\nWhen a thin viscoelastic coating is added to the sphere, the Rayleigh mode\nbehaviour is recovered in a limited frequency range, allowing the generation of\na collimating wave at the interface between the sphere and the coating.\n