Michael J. Cloud

The Rayleigh and Courant variational principles in the six-parameter shell theory

Courant’s minimax variational principle is considered in application to the six-parameter theory of prestressed shells. The equations of a prestressed micropolar shell are deduced in detail. Courant’s principle is used to study the dependence of the least and higher eigenfrequencies on shell parameters and boundary conditions. Cases involving boundary reinforcements and shell junctions are also treated.

On the natural frequencies of an annular ring above a conducting half space

The natural resonant frequencies of an object embedded in a stratified conducting medium are considered. The special case of an annular ring is considered in detail. Extensive numerical results are examined for the case of a ring in free space above a conducting half space. This simple problem provides insight into the behavior of the natural frequencies of airborne radar targets above the earth, ice, or sea surface. Results reveal an intricate pattern of pole trajectories in the complex plane. These trajectories are studied for various combinations of height, ring geometry, constitutive parameters, and modal index.

Natural Frequencies of a Conducting Sphere With a Circular Aperture

The natural frequencies of a hollow, perfectly conducting sphere with a circular aperture are calculated. The trajectories of the poles in the complex plane reveal an intriguing behavior which depends on whether the poles originate from internal or external sphere resonances. It is found that the modal degeneracy of the complete sphere is removed by the addition of the aperture.

Automatic Error Analysis Using Intervals

A technique for automatic error analysis using interval mathematics is introduced. A comparison to standard error propagation methods shows that in cases involving complicated formulas, the interval approach gives comparable error estimates with much less effort. Several examples are considered, and numerical errors are computed using the INTLAB extension to MATLAB. Two laboratory experiments are examined, and measured data are used to explore the applicability of interval analysis under typical experimental conditions. Because of the simplicity of using interval analysis, and because of its easy implementation in MATLAB, error analysis may be introduced in the earliest electrical engineering lab classes. This provides students with a crucial skill that will be valuable throughout their engineering studies.

Introduction to mathematical elasticity

Models and Ideas of Classical Mechanics Simple Elastic Models Theory of Elasticity: Statics and Dynamics.

A Hallen-type integral equation for symmetric scattering from lossy circular disks

A rigorous technique is presented for calculating the current induced on a thin lossy disk by rotationally symmetric sources, and the resulting scattered field. A Hallen-type integral equation is developed for the current using the magnetic vector potential, and it is solved by the method of moments. It is shown that the diffraction lobes usually associated with radiation above a finite circular ground plane can be reduced dramatically by the addition of loss. Application to a quarter-wave monopole radiating above a finite circular perfectly conducting ground plane shows good agreement with experiment. >

Elements of Nonlinear Functional Analysis

From the viewpoint of functional analysis, nonlinear problems of mechanics are more complicated than linear problems; as in mechanics, they require new techniques for their study. Many of them, such as nonlinear elasticity in the general case, provide a wide field of investigation for mathematicians (see Antman [2]) ; the problem of existence of solutions in nonlinear elasticity in general is still open.

A Brief Introduction to Interval Analysis

Some major advances in mathematics have occurred through the extension of existing number systems. The natural numbers were extended to the real numbers, the real numbers to the complex numbers, and so on.

Basic Review and Elementary Facts

Inequalities lie at the heart of mathematical analysis. They appear in the definitions of continuity and limit (and hence in the definitions of the integral and the derivative). They play crucial roles in generalizing the notions of distance and vector magnitude. But many problems of physical interest also rely on simple inequality concepts for their solution. In engineering, it is not always best to think in terms of equality. Let us illustrate this statement with a few examples.

Methods from the Calculus

In this chapter we revisit some facts from mathematical analysis and show how these may be used to establish important inequalities. We begin by reviewing convergence of real number sequences and continuity of real functions of a single variable.