9.1 INTRODUCTION [LO1]

Stresses given by relatively simple equations in the strength of materials for structures or machine members are based on the assumed continuity of the elastic medium. However, the presence of discontinuity destroys the assumed regularity of stress distribution in a member and a sudden increase in stresses occurs in the neighborhood of the discontinuity. In developing machines, it is impossible to avoid abrupt changes in cross-sections, holes, notches, shoulders, etc. Abrupt changes in cross-section also occur at the roots of gear teeth and threads of bolts. Some examples are shown in Figure 9.1.

Any such discontinuity acts as a stress raiser. Ideally, discontinuity in materials such as non-metallic inclusions in metals, casting defects, residual stresses from welding may also act as stress raisers. In this chapter, however, we shall consider only the geometric discontinuity that arises from design considerations of structures or machine parts.

Many theoretical methods and experimental techniques have been developed to determine stress concentrations in different structural and mechanical systems. In order to understand the concept, we shall begin with a plate with a centrally located hole. The plate is subjected to uniformly distributed tensile loading at the ends, as shown in Figure 9.2.

Introduction

All metals and alloys exhibit a reduction in electrical resistance as they cool. As the temperature drops, atoms’ thermal vibrations become less intense, and conduction electrons scatter less frequently. The resistivity should decrease toward zero as the temperature approaches zero Kelvin for a perfect pure metal, where the only thing standing in the way of an electron's travel is the thermal vibrations of the lattice. This zero resistance, which a hypothetical perfect specimen would acquire if it could be cooled to absolute zero, is the phenomenon of superconductivity. Any real specimen of metal cannot be perfectly pure and will contain some impurities. As a result, in addition to being scattered by the thermal vibrations of the lattice atoms, the electrons are also dispersed by impurities, and this impurity scattering is largely temperature independent. As a result, at the lowest temperature, there will be some residual resistance. The residual resistivity of a metal increases with the degree of impurity.

The phenomenon of superconductivity was first discovered by Dutch physicist H. Kamerling Onnes of Leiden University in 1911 during the investigation of the variation of electrical resistance of mercury in the newly available range of low temperatures, in the neighborhood of temperature of liquid helium (or 4.2 K). He observed that the resistance of mercury suddenly falls from 0.08 ohm at about 4 K to less than 3 × 10−6 ohm over a very small temperature of 0.01 K.

Introduction

The nonconducting materials such as paper, wood, glass, ceramics, polymers and so on do not have free charge carriers, that is, electrons or holes. Therefore, they prevent the flow of electrical current and heat through them.

When the main function of nonconducting materials is to provide electrical isolation then they are called insulators.

When the main function of nonconducting materials is for charge storage then it is called dielectric.

The dielectrics are polarized under the influence of an external electric field.

Dielectric Constant

Let us consider two parallel plates separated by a distance “d” connected with a dc supply of voltage V, as shown in Figure 6.1(a). Now the circuit is disconnected, and the dielectric is inserted between the plates, as shown in Figure 6.1(b).

Then, the voltage across the capacitor is reduced from V to V′. The change in voltage across the plates can be related by a factor as

Since V < V , the relative permittivity or dielectric constant ɛr 1 >.

The capacitance without dielectric is given as

The capacitance with dielectric is given as

Now, put the value of C and C¢ in equation (6.1), the relative permittivity or dielectric constant is

If “A” is the area of the plates, then

where ɛ is the permittivity of the material.

Introduction

In the early days, an operational amplifier (op-amp) was the only linear integrated circuit (IC) that was used in the design of linear IC circuits and systems. Typical applications of the op-amps were mathematical operations, such as summation, subtraction, integration, small signal amplification, and generating oscillations. Over the years, other devices, such as operational transconductance amplifiers, current conveyors, and so on, have also come into common use; still, it has not reduced the importance and areas of application of op-amps. Rather, it became possible to realize many more advanced functions with linear ICs and many applications coming under the domain of nonlinear applications with advances in the process technology and increased level of integration. Some of the more common nonlinear applications are precision rectifiers, voltage-level detectors, and Schmitt trigger circuits. The Schmitt trigger circuit itself is very popular in generating varieties of pulses and other waveforms like triangular waveforms. Some other nonlinear applications such as log and antilog amplifiers, analog multiplier, charge amplifier, and isolation amplifiers are discussed in brief; phase lock loop and its basic function are also included.

Precision Rectifiers

Conventional rectifiers work well for converting alternating supply to a pulsating one. Filters are normally used to remove ripples in the pulsating voltage to obtain dc. It is observed that these rectifiers have some limitations. One of the main limitations is that when a diode conducts during rectification, it has a voltage drop across its terminals, which is approximately 0.7 V. Hence, the ac voltage available for conversion to dc is reduced by that amount.

1.1 INTRODUCTION [LO1]

Mechanics is one of the oldest physical sciences, dating back to the times of Aristotle and Archimedes. The subject deals with force, displacement, and motion. The concepts of mechanics have been used to solve many mechanical and structural engineering problems through the ages. Because of its intriguing nature, many great scientists including Sir Isaac Newton and Albert Einstein delved into it for solving intricate problems in their own fields.

Engineering mechanics and mechanics of materials developed over centuries with a few experiment-based postulates and assumptions, particularly to solve engineering problems in designing machines and structural parts. Problems are many and varied. However, in most cases, the requirement is to ensure sufficient strength, stiffness, and stability of the components, and eventually those of the whole machine or structure. In order to do this, we first analyze the forces and stresses at different points in a member, and then select materials of known strength and deformation behavior, to withstand the stress distribution with tolerable deformation and stability limits. The methodology has now developed to the extent of coding that takes into account the whole field stress, strain, deformation behaviors, and material characteristics to predict the probability of failure of a component at the weakest point. Inputs from the theory of elasticity and plasticity, mathematical and computational techniques, material science, and many other branches of science are needed to develop such sophisticated coding.

The theory of elasticity too developed but as an applied mathematics topic, and engineers took very little notice of it until recently, when critical analyses of components in high-speed machinery, vehicles, aerospace technology, and many other applications became necessary. The types of problems considered in both the elementary strength of material and the theory of elasticity are similar, but the approaches are different. The strength of the materials approach is generally simple. Here the emphasis is on finding practical solutions to a problem with simplifying assumptions.

Wave optics is the branch of modern physics in which the nature of light and its propagation are studied.

Interference

When two waves of the same frequency, having a constant phase difference between them, and traveling in the same medium are allowed to superimpose each other, there is a modification in the intensity pattern. This phenomenon is known as interference of light.

When the resultant amplitude at certain points is the sum of the amplitudes of the two waves, this interference is known as constructive interference.

When the resultant amplitude at certain points is the difference of the amplitudes of the two waves, this interference is known as destructive interference, as shown in Figure 11.1.

COHERENT SOURCES

Two sources are said to be coherent if the waves emitted from them have a constant phase difference with time.

THEORY OF INTERFERENCE

Let us consider two coherent sources S1 and S2 that are equidistant from source S. Let a1 and a2 be the amplitudes of the waves originated from source S1 and S2, respectively, as shown in Figure 11.2. Then the displacement y1 from the source S is given by

where δ is the phase difference between the two waves.

Now, according to the law of superposition, the resultant wave is given by

Band Theory ofSolids

The band theory of solids is different from the others because the atoms are arranged very close to each other such that the energy levels of the outermost orbital electrons are affected. But the energy level of the innermost electrons is not affected by the neighboring atoms.

In general, if there is n number of atoms, then there will be n discrete energy levels in each energy band. In such a system of n number of atoms, the molecular orbitals are called energy bands shown in Figure 7.1.

CLASSIFICATION OF SOLIDS ON THE BASIS OF BAND THEORY

The solids can be classified on the basis of band theory. The parameter that differentiates the solids among insulator, conductor, and semiconductor is known as energy band gap and represented by (Eg), as shown in Figure 7.2. When the energy band gap (Eg) between conduction band and valence band is greater than 5 eV (electron-volt) then the solid is classified as insulator. When the energy band gap (E g)between conduction band and valence band is 0 eV (electron-volt), that is, overlapping of bands occurs then the solid is classified as conductor. When the energy band gap (Eg) between conduction band and valence band is approximately equals to 1 eV (electron-volt) then the solid is classified as semiconductors.

In this chapter, we shall study separation properties of topological spaces, vaguely speaking, whether two distinct points, a point and a closed set not containing the point, or two disjoint closed sets can be separated by disjoint open sets. On one extreme, we have indiscrete spaces in which nothing can be separated, and on the other end, we have discrete spaces in which every subset is open and it possesses all the separation properties. Metric spaces satisfy all these properties, and so, in a way, separation axioms attempt to seek how far is a topological space from being metrizable. Initially, the definition of topology given by Felix Hausdorff included a condition that we now know as the T2-axiom or the Hausdorff property. Later on, separation axioms were studied under the name accessible spaces, which was introduced by Fréchet, now known as T1-spaces and also as Fréchet space. The T0-space was introduced by Kolmogorov, and T0-spaces are also called Kolmogorov spaces. In 1923, Heinrich Tietze introduced the notation Ti for these spaces, which comes from the German word “Trennungsaxiomen,” which means “separation axioms,” and it has nothing to do with his name.

In the realm of ring theory, polynomial rings emerge as indispensable algebraic structures, providing a rich and versatile framework for studying a wide array of mathematical concepts. At their core, polynomial rings serve as a natural extension of the familiar concept of polynomials in a single variable, offering a systematic way to explore algebraic expressions involving multiple variables. This chapter delves into the foundational aspects of polynomial rings, elucidating their construction, properties, and significance within the broader landscape of ring theory.

A polynomial ring is constructed by formalizing expressions involving indeterminates and coefficients, embodying a powerful algebraic structure that captures the essence of polynomial manipulation. The algebraic properties of polynomial rings are examined, their role as noncommutative rings is emphasized, and how they form a foundation for understanding diverse mathematical topics is examined. From polynomial factorization to the roots of polynomials, polynomial rings offer insights into the structure and behavior of rings, making them a cornerstone in the exploration of abstract algebra. Furthermore, the chapter will explore connections between polynomial rings and other algebraic structures, shedding light on their significance from the perspective of mathematical theory. Through this exploration, readers will gain a deeper appreciation for the elegance and applicability of polynomial rings in the context of ring theory.