Adaptive Structures: Dynamics and Control

Abstract

These reviews of books and other forms of information express the opinions of the individual reviewers and are not necessarily endorsed by the Editorial Board of this Journal.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 announ...

Automotive Applications of Active Vibration Control

“…In control engineering terms, the transfer function from the amplifier input to the (filtered) sensor output is the transfer function of the plant to be controlled (assuming linearity and time invariance). In accordance with the active noise and vibration control literature (Kuo & Morgan, 1996), (Hansen & Snyder, 1997), (Clark et al, 1998) this is called the secondary path S. To design a control algorithm, a model for the secondary path is required. Quite often models for vibration control systems are derived from physical principles (Preumont, 1997), from finite-element models or through experimental techniques such as modal analysis (Heylen et al, 1997).…”

“…For an overview see e.g. (Kuo & Morgan, 1996), (Hansen & Snyder, 1997), (Clark et al, 1998), (Elliot, 2001) and the references therein. The main part of the published literature makes use of adaptive feedforward structures.…”

“…Most of these approaches rely on feedforward control strategies (either pure feedforward or combined with feedback). The feedforward signal is either taken from an additional sensor (usually an accelerometer in active vibration control) or generated artificially from measurements of the fundamental disturbance frequency (Kuo & Morgan, 1996), (Hansen & Snyder, 1997), (Clark et al, 1998), (Elliot, 2001). Contrary to the major fields of application for active noise and vibration control (military and aircraft), the automotive sector is extremely sensitive to the costs of the overall system.…”

“…It is therefore desirable to use an approach that requires only one sensor. Also, most approaches rely on adaptive control strategies such as the filtered-x LMS algorithm (Kuo & Morgan, 1996), (Hansen & Snyder, 1997), (Clark et al, 1998), (Elliot, 2001). This seems necessary as the characteristics of the disturbance acting upon the system are time varying.…”

Optimal control of piezoceramic actuators

“…It is based on the representation of the system and the control as finite impulse response filters (FIR), the coefficients of which are determined by least mean square algorithms. A detailed overview of its application to the control of smart structures is given in [1], see [2] and [3] for a general theory of adaptive filters and design. The method, however, is restricted to linear systems or systems that can be linearized and thus is only applicable in the low drive level range.…”