There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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

Shape-Memory Materials

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Seismic Design Of Reinforced Concrete And Masonry Buildings

Free vibration analysis of axially functionally graded tapered Bernoulli–Euler microbeams based on the modified couple stress theory

In this article, free vibration characteristics of rotating axially functionally graded tapered beams with six different boundary conditions are studied through a finite-element approach. The beam element employed in this study is a two-noded element whose shape functions are obtained in terms of special functions namely basic displacement functions. Application of this beam element provides good results as the shape functions take the effects of variable centrifugal force, cross-sectional area, and mechanical properties of the material. In this study, the effects of material non-homogeneity, taper ratio, rotating speed parameter, hub radius, and tip mass are investigated. The results are given in tabular form so that they could be used as a means of comparison for future studies in this field.

Free vibration of rotating axially functionally graded tapered beams

This paper deals with enhancing the existing Finite Element formulations through employing basic principles of structural mechanics accompanied with mathematical techniques. Introducing the concept of Basic Displacement Functions (BDFs), the free vibration analysis of rotating tapered beams is studied from a mechanical point of view. It is shown that exact shape functions could be derived in terms of BDFs. The new shape functions turn out to be dependent on the rotational speed, circular frequency, the position of element along the beam and variation of cross-sectional dimensions along the element. Dynamic BDFs are obtained by applying Adomian Modified Decomposition Method (AMDM) to the governing differential equation of motion. Carrying out numerical examples, the competency of the method is verified.

Dynamic basic displacement functions in free vibration analysis of centrifugally stiffened tapered beams; a mechanical solution

Rotating beams are considerably used in different mechanical and aeronautical installations. In this paper, free vibration of non-prismatic rotating Euler-Bernoulli beams is studied. Dynamic stiffness matrix is evaluated by using differential transform method, a powerful numerical tool in solution of ordinary differential equations, for solving the governing equation of motion. The method is capable of modeling any beam whose cross-sectional area and moment of inertia vary along beam with any two arbitrary functions and any type of cross-section with just one or few elements so that it can be used in most of engineering applications. In order to verify the competency of the method, natural frequencies are obtained for two problems and the effects of rotational speed parameter and taper ratio on natural frequencies are investigated.

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Application of Differential Transform Method in Free Vibration Analysis of Rotating Non-Prismatic Beams

Introducing the concept of Basic Displacement Functions (BDFs), an innovative method is presented which yields a mechanical based approach rather than a mathematical one for exact static analysis of arbitrarily tapered Timoshenko beams. Holding pure mechanical interpretations, BDFs are obtained using energy methods such as unit load method. It is shown that exact shape functions and consequently structural matrices could be derived in terms of BDFs. Unlike the most of finite element formulations which are based on prescribed displacement fields, the present finite element takes advantage of a flexibility basis which guarantees the exact interpolation of displacement field along the element and leads to fast convergence of the method in static analysis even with few elements. The method could be easily implemented into any standard displacement-based program. Carrying out numerical examples, the competency of the method in both static and free vibration is verified.

Analysis of Non-Prismatic Timoshenko Beams Using Basic Displacement Functions

This paper aims at extending the application of two-dimensional differential transform method (2D-DTM) to study the free vibration of thin plates with arbitrarily varying thickness. First, the differential equation of motion governing thin plates with varying thickness is derived using Hamilton’s principle. Afterward, the 2D-DTM, a numerical method which is capable of reducing the size of computational work and can be applied to various types of differential equations, has been applied to derive the natural frequencies of variable thickness thin plates with different boundary conditions. Several numerical examples have been carried out to demonstrate the applicability and accuracy of the present method in free vibration analysis of both uniform plates and plates with variable thickness.

Reza Attarnejad

Papers

Free vibration of rotating axially functionally graded tapered beams

Dynamic basic displacement functions in free vibration analysis of centrifugally stiffened tapered beams; a mechanical solution

Application of Differential Transform Method in Free Vibration Analysis of Rotating Non-Prismatic Beams

Analysis of Non-Prismatic Timoshenko Beams Using Basic Displacement Functions

Free vibration analysis of variable thickness thin plates by two-dimensional differential transform method