Adaptive Structures: Dynamics and Control
Robert L. Clark, William R. Saunders, Gary P. Gibbs, Scott D. Sommerfeldt · The Journal of the Acoustical Society of America · 2001
Dynamics of shells, plates, and rods is an engineering problem, which involves complicated mathematics due to the fourth-order partial differential equations that describe flexural vibrations.As a result there are two types of books related to this problem: one is written by engineers or physicists and the other by mathematicians.The reviewed book is clearly written by a mathematician, and, in my view, is written for mathematicians who specialize in mechanics and structural dynamics.This book is based on a hybrid variational-asymptotic approach that reduces the three-dimensional equations of elastic theory of plates, shells, and rods to two-and one-dimensional approximate equations of motion using small parameters.This variational-asymptotic ͑VA͒ method was developed by Professor V. L. Berdichevsky about 20 years ago studying small amplitude long-wave vibrations.The method is a synthesis of two methods that are widely used in mechanics: asymptotic and variational.The VA method starts by approximating the system Lagrangian of the functional, instead of a system of differential equations.Neglecting a small term in the Lagrangian is equivalent to neglecting several terms in the differential equations, which are not always easy to recognize as small ones.Next, approximate differential equations are obtained by varying the approximated functional.This approach simplifies the analysis and provides greater flexibility for modifying and solving the equations.The author of the book, Professor Le, has extrapolated this method into the high-frequency short-wave range.Specifically, he applies the VA method to analyze vibrations of piezoelectric shells, plates, and rods, and devotes roughly half of the book to this topic.The book starts with two introductory chapters that discuss the historical and mathematical background of tensor analysis, the geometry of curves and surfaces, the dynamic theory of elasticity and piezoelectricity, and the variational-asymptotic method.After this introduction the book is divided into two equal parts: the first part describes low-frequency vibrations and the second high-frequency vibrations.Both parts contain four chapters with the same titles: Elastic Shells, Elastic Rods, Piezoelectric Shells, and Piezoelectric Rods.Each section ends with problems and exercises.The bibliography consists of 62 references mostly related to the methods described in the book.Overall, the book describes an interesting and powerful method of deriving and solving complicated equations of shell and rod vibrations.However, as a physicist and practitioner, I hestitate to recommend it to ''engineers who deal with vibrations of shells and rods in their everyday practice.''I would rather agree that it is ''for mathematicians who seek applications of the variational and asymptotic methods in elasticity and piezoelectricity,'' as announced on the back cover of the book.