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This book focuses on the justification and refinement of highly diverse approximate dynamic models for engineering structures arising in modern technology, including high-tech domains involving nano- and meta-materials. It proposes a classification for vibration spectra over a broad frequency domain and evaluates the range of validity of various existing 2D theories for thin-walled shells by comparing them with 3D benchmark solutions. The dynamic equations in 3D elasticity are applied to the analysis of harmonic vibrations in hollow bodies with canonical shapes. New exact homogeneous and inhomogeneous solutions are derived for cylinders, spheres and cones (including spherical and conical layers), as well as for plates of variable thickness. The book includes a wealth of numerical examples, as well as refined versions of 2D dynamic formulations. Boundary value problems for hollow bodies are also addressed.

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Introduction.- 3D equations of dynamic elasticity in orthogonal co-ordinates.- Exact homogeneous and inhomogeneous solutions.- Cylinder of finite length.- Spherical layer.- Truncated cone .- Plates of variable thickness.- Free vibrations of cylinders and spheres.- Asymptotic analysis of thin-walled structures.- Validation of 2D engineering theories.- Conclusions.

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Honored Science Worker, academician Mekhtiyev Magomed Ferman oghlu graduated from department of Mechanics-Mathematics of Baku State University. He defended his Ph.D. thesis at the chair of elasticity theory of Rostov State University and there. In 1989 he defended his Doctoral thesis in Leningrad (St.Petersburg) State University. In 1966-1991 he occupied various positions in the Institute of Mechanics and Mathematics of National Academy of Sciences of Azerbaijan. He has worked in Baku State University since 1991. In 1994 he became professor. Scientific-research direction of prof. M.F.Mekhtiyev is mathematical methods of Solid Mechanics and Qualitative questions of optimal control. He has published over 120 scientific papers and two monographs in this field. M.F.Mekhtiyev is awarded with Gold medal of Scientific-industrial Chamber of European Union. At present he is Dean of the department of Applied Mathematics and Cybernetics and heads the chair of Mathematical Methods of Applied Analysis.

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