用一维半解析有限元法研究多层固体球中的受迫振动和波的传播
Forced vibrations and wave propagation in multilayered solid spheres using a one-dimensional semi-analytical finite element method
摘要：
本文提出了一种计算多层弹性球本征模态和受迫响应的数值模型。其主要思想是沿角坐标用球谐函数解析地描述问题，径向用一维有限元离散。必须仔细选择适当的测试函数，以便可以使用矢量和张量球谐正交关系。所提出的方法产生了一个一般的一维配方的角度行为的完全解析描述，适合于任何插值技术。一个线性特征值问题，简单，快速解决，然后得到。本征解是球振型和扭转振型。它们是有利的文献结果相比，一个均匀的球。将特征解叠加，显式计算强迫响应。后者用于重建表面波的传播。特别地，利用该模型恢复了均匀球体中线源激发的Rayleigh波（准等宽传播的非衍射表面波）的准直性。基于振动本征模，模态分析表明，这样的波是一个基本的球面模式的叠加与一个位移限制在赤道的球。这些模式是所谓的瑞利模式，扇区类型和高极性波数。当一个薄的粘弹性涂层被添加到球体，瑞利模式的行为是在有限的频率范围内恢复，允许在球体和涂层之间的界面处的准直波的产生。
英文摘要：
A numerical model is proposed to compute the eigenmodes and the forced response of multilayered elastic spheres. The main idea is to describe analytically the problem along the angular coordinates with spherical harmonics and to discretize the radial direction with one-dimensional finite elements. The proper test function must be carefully chosen so that both vector and tensor spherical harmonics orthogonality relationships can be used. The proposed approach yields a general one-dimensional formulation with a fully analytical description of the angular behaviour, suitable for any interpolating technique. A linear eigenvalue problem, simple and fast to solve, is then obtained. The eigensolutions are the spheroidal and torsional modes. They are favourably compared with literature results for a homogeneous sphere. The eigensolutions are superposed to compute explicitly the forced response. The latter is used to reconstruct the propagation of surfaces waves. In particular, the collimation of a Rayleigh wave (non-diffracted surface wave propagating with a quasi-constant width) excited by a line source in a homogeneous sphere is recovered with the model. Based on the vibration eigenmodes, a modal analysis shows that such a wave is a superposition of fundamental spheroidal modes with a displacement confined at the equator of the sphere. These modes are the so-called Rayleigh modes, of sectoral type and high polar wavenumbers. When a thin viscoelastic coating is added to the sphere, the Rayleigh mode behaviour is recovered in a limited frequency range, allowing the generation of a collimating wave at the interface between the sphere and the coating.
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