This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Introduction to the Finite Element Method

THE appearance of a treatise in English upon the mathematical theory of elasticity is an event the potential importance of which may be judged by the that the author, in his frequent suggestions for collateral reading, refers to only three such, those of Southwell, Timoshenko, and Love. In spirit and content Sokolnikoff}s book differs greatly from each and all of these. It may be described by a possible sub-title: “A pure mathematician surveys topics related to certain problems in the mathematical theory of elasticity”. It is symptomatic of the change in outlook of American mathematics over the past few decades. Mathematical Theory Of Elasticity Prof. I. S. Sokolnikoff with the collaboration of Asst. Prof. R. D. Speche. Pp. xi + 373. (New York and London: McGraw-Hill Book Co., Inc., 1946.) 22s. 6d.

Mathematical Theory Of Elasticity

1.Introduction to Composite Materials 2. Macrochemical Behavior of a Lamina 3.Micromechanical Behavior of a Lamina 4.Macromechanical Behavior of a Laminate 5.Bending, Buckling, and Vibration of Laminated Plates 6.Other Analysis and Behavior Topics 7.Introduction to Design of Composite Structures Appendix A.Matrices and Tensors Appendix B.Maxima and Minima of Functions of a Single Variable Appendix C.Typical Stress-Strain Curves Appendix D.Governing Equations for Beam Equilibrium and Plate Equilibrium, Buckling, and Vibration Index

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Mechanics of composite materials

Mathematical models of elastic structures have become very sophisticated: given the crucial material properties (mass density and the several elastic moduli), computer-based techniques can be used to construct exotic finite element models. By contrast, the modeling of damping is usually very primitive, often consisting of no more than mere guesses at “modal damping factors.” The aim of this paper is to raise the modeling of viscoelastic structures to a level consistent with the modeling of elastic structures. Appropriate material properties are identified which permit the standard finite element formulations used for undamped structures to be extended to viscoelastic structures. Through the use of “dissipation” coordinates, the canonical “M , K ” form of the undamped motion equations is expanded to encompass viscoelastic damping. With this formulation finite element analysis can be used to model viscoelastic damping accurately.

Dynamics of viscoelastic structures: a time-domain finite element formulation

A. M. Freudenthal London: John Wiley. 1966. Pp. xv + 492. Price £5 13s. Although written in modern idiom with a currently fashionable title, the book deals with a subject formerly called 'Strength of materials', and covers what engineers are expected to know about stresses in simple structures and mechanical properties of materials. The Introduction admirably discusses how the theoretical and experimental approaches to the behaviour of mechanical systems are to be reconciled with each other.

Introduction to the Mechanics of Solids

Active vibration control of smart piezoelectric beams: Comparison of classical and optimal feedback control strategies

Arbitrary active constrained layer damping treatments on beams: Finite element modelling and experimental validation

This paper presents a theoretical and finite element (FE) formulation of a three-layered smart beam with two piezoelectric layers acting as sensors or actuators. For the definition of the mechanical model a partial layerwise theory is considered for the approximation of the displacement field of the core and piezoelectric face layers. An electrical model for different electric boundary conditions (EBC), namely, electroded layers with either closed- or open-circuit electrodes with electric potential prescribed or layers without electrodes, is considered. Using a variational formulation, the direct piezoelectric effect for the different EBC is physically incorporated into the mechanical model through appropriate approximations of the electric field in the axial and transverse directions. An FE model of a three-layered smart beam with different EBC is proposed considering a fully coupled electro-mechanical theory through the use of effective stiffness parameters and a modified static condensation. FE solutions of the quasi-static electrical and mechanical actuations and natural frequencies are presented. Comparisons with numerical FE and analytical solutions available in the literature demonstrate the representativeness of the developed theory and the effectiveness of the proposed FE model for different EBC.

Coupled three‐layered analysis of smart piezoelectric beams with different electric boundary conditions

Combined feedback/feedforward active control of vibration of beams with ACLD treatments: Numerical simulation

A velocity feedback control system is evaluated in the active control of vibrations of a smart beam with a pair of surface mounted piezoelectric ceramic patches, and finite element (FE) model results are validated against measured ones. To this end, a three-layered smart beam FE model is utilized, where a partial layerwise theory and a fully coupled electro-mechanical theory are considered for the formulation of the displacement field and electric potential, respectively. Regarding the test rig, it consists of a cantilever smart aluminum beam with two piezoelectric patches mounted close to the clamped end. One of the piezoelectric patches is utilized to excite the beam while the other is utilized as an actuator in the feedback control loop. The control voltage applied to the actuator is proportional to the transverse velocity at the free end, which is measured by a laser vibrometer. First, the quasi-static actuation capacity of the piezoelectric patches is evaluated. Next, the free and forced velocity responses to an initial displacement field and harmonic excitation are analyzed. The capacity to predict instabilities and the accuracy of the FE model are demonstrated and the applicability and functionality of the velocity feedback vibration control system are discussed.

https://iopscience.iop.org/article/10.1088/0964-1726/16/2/008/pdf

Active vibration control of a smart beam through piezoelectric actuation and laser vibrometer sensing: simulation, design and experimental implementation

C. M. A. Vasques

Papers

Active vibration control of smart piezoelectric beams: Comparison of classical and optimal feedback control strategies

Arbitrary active constrained layer damping treatments on beams: Finite element modelling and experimental validation

Coupled three‐layered analysis of smart piezoelectric beams with different electric boundary conditions

Combined feedback/feedforward active control of vibration of beams with ACLD treatments: Numerical simulation

Active vibration control of a smart beam through piezoelectric actuation and laser vibrometer sensing: simulation, design and experimental implementation