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

This study investigates the response frequency conversion characteristic of a nonlinear curved panel mounted with a centre mass and the sound radiations. A set of coupled governing differential equations is set up and used to generate the nonlinear vibration responses, which are used to calculate the corresponding radiated sounds. The vibration, sound levels, and the ratio of the antisymmetrical to symmetrical mode responses are plotted against the excitation level and compared with a set of experimental data. The frequency conversion characteristic is investigated from the frequency spectrums of the vibration responses.

/pdf/the-response-frequency-conversion-characteristic-of-a-55fzmihvk4.pdf

The Response Frequency Conversion Characteristic of a Nonlinear Curved Panel with a Centre Mass and the Sound Radiations

This paper addresses the analysis for the nonlinear vibration response of a rectangular tube with a flexible end and non-rigid acoustic boundaries. The structural–acoustic modal formulations are developed from the Duffing differential equation and wave equation, which represent the large-amplitude structural vibration of a flexible panel coupled with a cavity. This problem considers both non-rigid acoustic boundary and structural cubic nonlinearity. The multi-level residue harmonic balance method is employed for solving the nonlinear coupled differential equations developed in the problem. The results obtained from the proposed method and numerical method are generally in good agreement. The effects of excitation magnitude, tube length, and phase shift parameter, etc., are examined.

/pdf/nonlinear-vibration-response-of-a-rectangular-tube-with-a-2wwwjupecr.pdf

Nonlinear Vibration Response of a Rectangular Tube with a Flexible End and Non-Rigid Acoustic Boundaries

This paper addresses the effect of leakage on the natural frequencies of a large amplitude vibrating panel backed by a cavity, which has not been considered in many other related studies. The structural-acoustic governing equations are employed to study this nonlinear problem. An elliptical integral method, which was recently developed for the nonlinear panel cavity problem, is introduced here to solve for the structural-acoustics responses. The present results agree reasonably well with those obtained from the classical harmonic balance method. Modal convergences of the nonlinear solutions are performed to verify the proposed method. The effects of vibration amplitude and leakage size are studied and discussed. It is found that (1) the edge leakages in a panel cavity system significantly affect the natural frequency properties, and (2) the edge leakages induce a low frequency acoustic resonance.

Large amplitude free vibration of a flexible panel coupled with a leaking cavity

This study addresses the nonlinear structure-extended cavity interaction simulation using a new version of the multilevel residue harmonic balance method. This method has only been adopted once to solve a nonlinear beam problem. This is the first study to use this method to solve a nonlinear structural acoustic problem. This study has two focuses: 1) the new version of the multilevel residue harmonic balance method can generate the higher-level nonlinear solutions ignored in the previous version and 2) the effect of the extended cavity, which has not been considered in previous studies, is examined. The cavity length of a panel-cavity system is sometimes longer than the panel length. However, many studies have adopted a model in which the cavity length is equal to the panel length. The effects of excitation magnitude, cavity depth, damping and number of structural modes on sound and vibration responses are investigated for various panel cases. In the simulations, the present harmonic balance solutions agree reasonably well with those obtained from the classical harmonic balance method. There are two important findings. First, the nonlinearity of a structural acoustic system highly depends on the cavity size. If the cavity size is smaller, the nonlinearity is higher. A large cavity volume implies a low stiffness or small acoustic pressure transmitted from the source panel to the nonlinear panel. In other words, the additional volume in an extended cavity affects the nonlinearity, sound and vibration responses of a structural acoustic system. Second, if an acoustic resonance couples with a structural resonance, nonlinearity is amplified and thus the insertion loss is adversely affected.

/pdf/nonlinear-structure-extended-cavity-interaction-simulation-4ylu8mqfp3.pdf

Nonlinear structure-extended cavity interaction simulation using a new version of harmonic balance method.

In this paper, a close-fitting enclosure model is used to prove the existence of a common structural-acoustic coupling experimentally, which deteriorates the insertion loss performance but was not considered in the formulas developed by Refs. (1) - ( 3) and the classical formula used in Refs. (4) - (6). The experimental results show that an enclosure, which first anti-symmetric structural and acoustic resonant frequencies do not coincide or nearly coincide, can avoid strong structural-acoustic couplings within the air cavity. At the acoustic resonance frequencies, the insertion loss performance of an enclosure can be improved by using the aluminum fiberboard, porous aluminum, and perforated aluminum panel to absorb the sound energy. Stiffener is useful to enhance the insertion loss ability of an enclosure if the dominant frequency bands of the noise are below the first structural resonant frequency. For having good damping effect to suppress the noise radiation at structural resonant frequencies, an enclosure should be mounted with both stiffeners and damping treatments.

Yiu-Yin Lee

Papers

The Response Frequency Conversion Characteristic of a Nonlinear Curved Panel with a Centre Mass and the Sound Radiations

Nonlinear Vibration Response of a Rectangular Tube with a Flexible End and Non-Rigid Acoustic Boundaries

Large amplitude free vibration of a flexible panel coupled with a leaking cavity

Nonlinear structure-extended cavity interaction simulation using a new version of harmonic balance method.

Noise reduction techniques for close-fitting enclosures at structural/acoustic resonance