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

http://ieeexplore.ieee.org/iel5/5/31047/01444179.pdf

Fundamentals of acoustics

Damage identification in beam-type structures: frequency-based method vs mode-shape-based method

This paper is concerned with the free vibration problem for micro/nanobeams modelled after Eringen's nonlocal elasticity theory and Timoshenko beam theory. The small scale effect is taken into consideration in the former theory while the effects of transverse shear deformation and rotary inertia are accounted for in the latter theory. The governing equations and the boundary conditions are derived using Hamilton's principle. These equations are solved analytically for the vibration frequencies of beams with various end conditions. The vibration solutions obtained provide a better representation of the vibration behaviour of short, stubby, micro/nanobeams where the effects of small scale, transverse shear deformation and rotary inertia are significant. The exact vibration solutions should serve as benchmark results for verifying numerically obtained solutions based on other beam models and solution techniques.

http://iopscience.iop.org/article/10.1088/0957-4484/18/10/105401/pdf

Vibration of nonlocal Timoshenko beams

In this paper, an efficient and simple higher order shear and normal deformation theory is presented for functionally graded material (FGM) plates. By dividing the transverse displacement into bending, shear and thickness stretching parts, the number of unknowns and governing equations for the present theory is reduced, significantly facilitating engineering analysis. Indeed, the number of unknown functions involved in the present theory is only five, as opposed to six or even greater numbers in the case of other shear and normal deformation theories. The present theory accounts for both shear deformation and thickness stretching effects by a hyperbolic variation of all displacements across the thickness, and satisfies the stress-free boundary conditions on the upper and lower surfaces of the plate without requiring any shear correction factor. Equations of motion are derived from Hamilton’s principle. Analytical solutions for the bending and free vibration analysis are obtained for simply supported plates. The obtained results are compared with 3-dimensional and quasi-3-dimensional solutions and those predicted by other plate theories. It can be concluded that the present theory is not only accurate but also simple in predicting the bending and free vibration responses of functionally graded plates.

An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates

The local vibration modes due to impact on the edge of a viaduct

This paper investigates large amplitude multi-mode free vibration andrandom response of thin cylindrical panels of rectangular planform usinga finite element modal formulation. A thin laminated composite doublycurved element is developed. The system equation in structural nodal DOFis transformed into the modal coordinates by the using the modes of theunderlying linear system. The nonlinear stiffness matrices are alsotransformed into nonlinear modal stiffness matrices. Numericalintegration is employed to determine free vibration and random response.Single-mode free vibration results are compared with existing classicalanalytical solutions to validate the nonlinear modal formulation.Nonlinear random analysis results for cylindrical panels have shown thatthe root mean square of panel deflections could be larger than thoseobtained using the linear structure theory. Time histories, probabilitydistribution functions, power spectral densities, and phase plane plotsare also presented.

Nonlinear Random Response of Cylindrical Panels to Acoustic Excitations Using Finite Element Modal Method

Reconstruction of the interior sound pressure of a room using the probabilistic approach


 A study of the use of H∞ control design to suppress the vibrations of composites plates embedded with piezoelectric actuators under random loading is presented. The finite element method is employed to obtain the dynamic equations of a plate vibration of fully coupled structural and electrical nodal degrees of freedom (DOF). The modal reduction method is adopted to reduce the finite element equations into a set of modal equations with fewer DOF. The modal equations are then employed for controller design and simulation. Results of the modal convergence study of the closed loop systems show that the truncated modes, which are neglected in the control design, deteriorate the controller performance and affect the optimal location and size of the actuator. The vibrations of plates subjected under off-resonant narrow-band excitations cannot be suppressed effectively. The controller shaped with an extra weight function for mitigating measurement noise effect gives better vibration reduction performance.

Control of random vibrations of composite plates with piezoelectric actuators

This article analyzes nonlinear vibration of an axially moving beam subject to periodic lateral forces. The governing equation of the beam vibration is developed using Hamilton's Principle. The stable and unstable periodic solutions are obtained by employing the multivariable Floquet theory and incremental harmonic balance (IHB) method. In the solution procedure, Hsu's method is applied for computing the transition matrix at the end of one period. The effects of internal resonance on the beam responses are discussed. The periodic solutions obtained from the IHB method are in good agreement with the results obtained from numerical integration.

Yiu-Yin Lee

Papers

The local vibration modes due to impact on the edge of a viaduct

Nonlinear Random Response of Cylindrical Panels to Acoustic Excitations Using Finite Element Modal Method

Reconstruction of the interior sound pressure of a room using the probabilistic approach

Control of random vibrations of composite plates with piezoelectric actuators

Nonlinear Analysis of Forced Responses of an Axially Moving Beam by Incremental Harmonic Balance Method.