chapter 15, Applications

from: Mathematical Techniques for Engineers and Scientists

Author(s): Larry C. Andrews , Ronald L. Phillips

Published: 22 April 2003

Chapter DOI: 10.1117/3.467443.ch15

Chapter Page Count: 50 pages

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Chapter Contents

15.1 Introduction
15.2 Mechanical Vibrations and Electric Circuits
15.3 Buckling of a Long Column
15.4 Communication Systems
15.5 Applications in Geometrical Optics
15.6 Wave Propagation in Free Space
15.7 ABCD Matrices for Paraxial Systems
15.8 Zernike Polynomials
Exercises

Excerpt

15.1 Introduction

In this final chapter we wish to present a collection of applications that make use of the various mathematical techniques introduced in the preceding chapters. For this purpose we have selected problems in mechanical vibrations, communication systems, and optics, among others.

15.2 Mechanical Vibrations and Electric Circuits

To begin, we consider some initial value problems (IVPs) involving DEs of the second order in connection with mechanical vibrations and electric circuits. Problems in both of these application areas are mathematically similar, the general problem being to solve the linear DE

where y′ = dy/dt, y″ = d²y/dt², subject to the prescribed initial conditions (ICs)

15.2.1 Forced oscillations—I

When a given weight (having mass m) is attached to an elastic spring suspended from a fixed support (like a ceiling beam), the spring will stretch to an equilibrium position by an amount s that varies with the weight mg, where g = 9.8 m/s² (32 ft/s²) is the gravitational constant. To remain in equilibrium, Hooke's law states that the spring will exert an upward restoring force f proportional to the amount of stretch; that is,

where the spring constant k depends on the “stiffness” of the spring. If the spring-mass system is also subjected to a downward (positive direction) external force F(t), the body or mass will move in the vertical direction. In addition to the external force, there may exist a retarding force caused by resistance of the medium in which the motion takes place, or possibly by friction. For example, the mass could be suspended in a viscous medium (like oil), connected to a dashpot damping device (like a shock absorber), and so on. In practice, many such retarding forces are approximately proportional to the velocity of the moving body and act in a direction opposing the motion.

As a consequence of Newton's second law of motion (F = ma), the sum of forces acting on the spring-mass system leads to the governing DE

where c is a positive retarding force constant and y represents the displacement from equilibrium of the mass alone. Such a spring-mass system is illustrated in Fig. 15.1.

DOI: 10.1117/3.467443.ch15

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