Oscillations of Elastic Bodies with Small Heavy Inclusions (Concentrated Masses)

We construct asymptotics of eigenfrequencies and eigenmodes of a composite anisotropic body with a group of small inclusions while mass of each is larger or equal in order the mass of the surrounding material. If a part of the body surface is rigidly clamped, modes of the natural oscillations are localized in main near the inclusions while the principal asymptotic terms of eigenfrequencies are described by the spectrum of problems about inclusions of unit size in the weightless space. In the case when the body surface is traction free, an interaction of small heavy inclusions is observed, namely the limiting problem consists of system of equations for inclusions in the space which are combined into a single spectral problem with integral terms at the spectral parameter. The structure of the integro-differential equations depends on the mass concentration coefficient as well as disposition of the inclusions. Justification of the derived asymptotic expansions is performed in a representable and most complicated case of superheavy concentrated masses distributed along a line and other situations are considered in the same way.