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...
This paper demonstrates active structural acoustic control using multiple input/output adaptive sensoriactuators combined with radiation filters and a feedback control paradigm. A new method of reduced order modeling/design of radiation filters termed radiation modal expansion (RME) is presented. For the experiments detailed in this paper, the RME technique reduced the modeling of the radiation matrix from 400 transfer functions to 6 transfer functions (multiplied by a constant transformation matrix). Experimental results demonstrate reductions of radiated sound power on the order of 5 dB over the bandwidth of 0-800 Hz.
In this paper, an approximate analytical model is developed for the excitation of a thin beam by a single piezoelectric actuator bonded to the surface of the beam. The premise of this work is to investigate the excitation of beams by piezoelectric actuators on a more fundamental level than present work, and then use the asymmetric model to predict a wave response, rather than a modal response, on more complicated structure/actuator systems. It is determined that the single surface mounted piezoelectric actuator simultaneously excites both flexural and
An experimental study of the application of discrete-time, linear quadratic control design methods to the cavity tone problem is described. State space models of the dynamics from a synthetic jet actuator at the leading edge of the cavity to two pressure sensors in the cavity were computed from experimental data. Variations in model order, control order, control bandwidth, and properties of a Kalman state estimator were studied. Feedback control reduced the levels of multiple cavity tones at Mach 0.275, 0.35, and 0.45. Closed loop performance was often limited by excitation of sidebands of cavity tones, and creation of new tones in the spectrum. State space models were useful for explaining some of these limitations, but were not able to account for non-linear dynamics, such as interactions between tones at different frequencies.
In this paper, an approximate analytical model is developed for the excitation of a thin beam by a single piezoelectric actuator bonded to the surface of the beam. The premise of this work is to investigate the excitation of beams by piezoelectric actuators on a more fundamental level than present work, and then use the asymmetric model to predict a wave response, rather than a modal response, on more complicated structure/actuator systems. It is determined that the single surface mounted piezoelectric actuator simultaneously excites both flexural and
This work addresses the design and application of robust controllers for structural acoustic control. Both simulation and experimental results are presented. H ∞ and µ-synthesis design methods were used to design feedback controllers which minimize power radiated from a panel while avoiding instability due to unmodeled dynamics. Specifically, highorder structural modes which couple strongly to the actuator-sensor path were poorly modeled. This model error was analytically bounded with an uncertainty model, which allowed controllers to be designed without artificial limits on control effort. It is found that robust control methods provide the control designer with physically meaningful parameters with which to tune control designs and can be very useful in determining limits of performance. Experimental results also showed, however, poor robustness properties for control designs with ad-hoc uncertainty models. The importance of quantifying and bounding model errors is discussed.
An experimental study of the application of discrete-time, linear quadratic control design methods to the cavity tone problem is described. State space models of the dynamics from a synthetic jet actuator at the leading edge of the cavity to two pressure sensors in the cavity were computed from experimental data. Variations in model order, control order, control bandwidth, and properties of a Kalman state estimator were studied. Feedback control reduced the levels of multiple cavity tones at Mach 0.275, 0.35, and 0.45. Closed loop performance was often limited by excitation of sidebands of cavity tones, and creation of new tones in the spectrum. State space models were useful for explaining some of these limitations, but were not able to account for non-linear dynamics, such as interactions between tones at different frequencies.
The simultaneous active control of flexural and extensional vibrations in elastic beams is experimentally investigated. The results demonstrate that using pairs of piezoceramic transducers, whose elements are symmetrically located and independently controlled by a multichannel adaptive controller, enables the high attenuation of both flexural and extensional response. This capability is due to the nature of the piezoceramic element, which when bonded to the surface of the structure and electrically excited, exerts a surface strain on the structure. This strain enables input of both shear forces and moments into the structural system. The results are applicable to many situations where extensional vibrations couple to large flexural vibrations and subsequently radiate significant sound levels.
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