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

Flow angle, temperature and aerodynamic damping on supersonic panel flutter stability boundary

一种针对有扰动项的耦合可积非色散方程的修正残差谐波平衡求解方法 本文将改进残余谐波平衡方法用于求解有扰动项的耦合可积非色散方程,并简化取得破解方案的过程。 1. 在取得每一阶段破解方案的过程中, 只需处理一条非线性代数方程式及一组线性代数方程式;2. 能找出旧方法不能找出的非线性答案。 1. 使用理论推导、方程式替换及残余谐波平衡方法;2. 通过仿真模拟,推导震动位移与频率之间的关系(图1)以及位移与速度之间的关系(图2)。 1. 成功将改进残余谐波平衡方法应用于有扰动项的耦合可积非色散方程;2. 通过与其他方法产生的数据进行比较,验证了所提方法的可行性和有效性(表1–3)。

Modified residue harmonic balance solution for coupled integrable dispersionless equations with disturbance terms

The effects of local modes on vibration reduction of building isolation

This study is the first to investigate the effect of chaotic vibration on the sound absorption of a curved perforated panel. Previous studies on the effect of nonlinear vibration on the sound absorption of a panel absorber have focused on periodic responses only. In this study, a sound absorption formula was derived by considering the panel impedance and perforation impedance. The numerical integration method was adopted to generate various chaotic vibrational responses, which were used to compute the corresponding sound absorptions. Several interesting findings that have never been observed in any previous studies on acoustic absorption were derived. First, in the chaotic and highly nonlinear cases, as the excitation frequency increased, the corresponding response frequencies decreased. This was opposite to the typical trend in linear cases, in which higher excitation frequencies corresponded to higher response frequencies. Second, in chaotic cases, absorption mainly occurred due to panel vibration effects. This is also in stark contrast to the findings of studies on perforated vibrating panels, in which the absorption effect mainly originates from perforations. Additionally, the absorption bandwidths are much wider and can shift to higher frequencies; however, the peak absorption coefficients were approximately 20% lower than in the case of the perforation effect only. Third, in the quasi-chaotic case, the absorption curve in the case of the perforation effect plus the vibration effect was between the absorption curves of the perforation effect only and the perforation effect plus the vibration effect.

Chaotic Vibration and Perforation Effects on the Sound Absorption of a Nonlinear Curved Panel Absorber

An analytical quadrilateral p element is developed for solving the free vibrations of piezoelectric-laminated plates. The formulations of the displacement and strain fields are based on first-order shear deformation plate theory. The coupling effect between the electrical and stress fields is also considered. The Legendre orthogonal polynomials are used as the element interpolation functions, and the analytical integration technique is adopted. It is found that the present p element method gives high numerical precision results, fast and monotonic convergence rate. In the numerical cases, the effects of the number of hierarchical terms and mesh size on the convergence rate are investigated. Examples of square plates with different displacement and potential boundary conditions are studied. In the comparisons, the solutions of the present element are in good agreement with those obtained from other classical and finite element methods.

https://downloads.hindawi.com/journals/aaa/2012/290461.pdf

Structural-Electrical-Coupled Formulation for the Free Vibration of a Piezoelectric-Laminated Plate Using the Analytical Arbitrary Quadrilateral p Element

Yiu-Yin Lee

Papers

Flow angle, temperature and aerodynamic damping on supersonic panel flutter stability boundary

Modified residue harmonic balance solution for coupled integrable dispersionless equations with disturbance terms

The effects of local modes on vibration reduction of building isolation

Chaotic Vibration and Perforation Effects on the Sound Absorption of a Nonlinear Curved Panel Absorber

Structural-Electrical-Coupled Formulation for the Free Vibration of a Piezoelectric-Laminated Plate Using the Analytical Arbitrary Quadrilateral p Element