Leonid M. Lyamshev
N. N. Andreev Acoustical Institute, Shvernik str. 4, 117036 Moscow, Russia
The oscillatory vibrations of inhomogeneous elastic structures are considered. In particular, two problems involving plates and shells with attached ribs excited by forces and moments are discussed. The formulation uses the Green's function for the wave equation of sound propagation in a liquid, and the equations of motion of plates and shells coupled with those of flexural and torsional vibrations of ribs. An integral relationship connecting the solutions of two self-conjugated boundary problems of inhomogeneous plates and shells oscillations theory is obtained. This relationship is the reciprocity theorem that is a precise mathematical formulation of the well-known Rayleigh reciprocity principle for oscillating mechanical systems. If a liquid is moving and there is a shear flow outside or inside of a shell (plate) then this relationship is the modified reciprocity theorem for inhomogeneous elastic structures oscillations in a flow that is the generalization of the Rayleigh principle. Applications of the reciprocity theorems are discussed. Flexural waves reflection, scattering, and radiation by elastic ribs in plates are considered. Nonspecular reflections and resonance phenomena are observed. These are similar to nonspecular reflection, sound scattering, and radiation by elastic bodies in a liquid and are connected with interactions of flexural waves in a plate with flexural and rotational waves in ribs.