This book is intended for engineers who deal with vibrations of rods and shells in their everyday practice but also wish to understand the subject from the mathematical point-of-view. Particularly the results concerning high-frequency vibrations may be new to them. The book can also serve as a textbook for advanced students. It may also be interesting for mathematicians who seek applications of the variational and asymptotic methods in elasticity and piezoelectricity. Only a minimum knowledge in advanced calculus and contiuum mechanics is assumed on the part of the reader.

We live in a world of vibrations and waves, without which there would not be sound, light, radio, television, communication etc. That is why the study of vibrations and waves is so important in many branches of physics and me- chanics. This book is devoted to the study of small mechanical vibrations of shells and rods, which are made of elastic or piezoelectric materials. But even in this very special field there are already many excellent books and mono- graphs written since the monumental work by Rayleigh [47]. The peculiarity of the present book is that we regard the equations of shells and rods as two- and one-dimensional approximate equations which can be derived from the three-dimensional theory by using the variational-asymptotic method. The latter has been invented especially for those variational problems which contain small parameters. It turns out that for vibrations of shells and rods there are many situations in which such small parameters exist. Thus, the application of the variational-asymptotic method enables one to derive not only the classical two- and one-dimensional theories of low-frequency vibra- tions of shells and rods, but also the theories of high-frequency (or thickness) vibrations.