Vibrations of Shells and Rods

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.