The introduction of high-speed electronic computers into engineering has opened the way for the solution of vibration problems of great complexity. One of the most suitable ways of expressing a problem for computational analysis is to use matrices. This book is concerned with the matrix formulation of the equations of motion of vibrating systems, and with techniques for the solution of matrix equations.

The purely computational side of practical vibration analysis naturally divides into two parts. The first is the construction of a mathematical model for the vibrating system, that is the setting up of equations governing its motion. The second is the process of solving the mathematical equations and extracting the properties of the solutions. This book is mainly concerned with the second part of the analysis. It will be assumed, on the whole, that the reader is acquainted with the various ways of setting up the equations of motion of vibrating systems.

This book can be regarded as a sequel to an earlier volume. The first, as its title suggests, is concerned mainly with the mechanics—the physical side—of vibration: this book is concerned more with the mathematical aspects of the subject. The connection between the two books is loose and they may be read independently of each other.

The fact that the book is concerned with the mathematical side of vibration theory, rather than the physical, throws emphasis on examples.

It is a fundamental fact of nature that the space we live in is threedimensional. Consequently, many branches of applied mathematics and theoretical physics are concerned with physical quantities defined in 3-space, as I shall call it; these subjects include Newtonian mechanics, fluid mechanics, theories of elasticity and plasticity, nonrelativistic quantum mechanics, and many parts of solid state physics. The Greek geometers made the first systematic investigation of the properties of ‘ordinary’ 3-space, and their work is known to us mainly through the books of Euclid; our basic geometrical ideas about the physical world have their origins in Euclidean geometry. A major advantage of Euclid's work was its presentation as a deductive system derived from a small number of definitions and axioms (or ‘basic assumptions’); although Euclid's axioms have turned out to be inadequate in a number of ways, he nevertheless provided us with a model of what a proper mathematical system should be [Reference p.i].

Through the introduction of coordinate systems, Descartes linked geometry with algebra [Reference P.2]; geometrical structures in 3-space such as lines, planes, circles, ellipses and spheres, were associated with algebraic equations involving three Cartesian coordinates (x, y, z). Then in the nineteenth century, Hamilton [Reference P.3] and Gibbs [Reference P.4] introduced two similar types of algebraic objects, ‘quaternions’ and ‘vectors’, which treated the three coordinates simultaneously; the rules of operation of these new sets of objects were different from those of real or complex numbers, giving rise to new types of ‘algebra’; a more general algebra of N-dimensional space (N = 3, 4, 5, …) was introduced by Grassmann [Reference P.5].

Introduction

Our understanding of the physical world depends to a great extent on making more or less exact measurements of a variety of physical quantities. All single measurements on a physical system consist of observing a single real number, and very often this single real number is, by itself, the value of an important physical quantity; examples are the measurement of a mass, a length, an interval of time, an electrical potential, the frequency or wavelength of an electromagnetic wave, a quantity of electrical charge, and the electric current in a wire. Physical quantities of this kind are called scalar quantities, or, more frequently, scalars. We shall make a distinction between these two expressions: ‘scalar’ will be used as a mathematical expression; scalars, for our purposes, are real algebraic variables λ, μ, …, which can, in general, take values in the whole range (-∞, ∞); they possess other properties which will be defined in Chapter 4, but for the present we shall regard them simply as real numbers. The expression ‘scalar quantity’ will refer to any specific physically measurable quantity, such as a mass or a charge, which is found experimentally to have the mathematical properties of a scalar. One important property of scalar quantities is that they are intrinsic properties of a physical system, and do not change if the whole physical system is translated to a different position in three-dimensional space, or is rotated in space.

Scalar and vector fields

In almost every branch of mathematical physics, we have to deal with physical quantities which extend continuously through regions of 3-space. These regions and their boundaries do not usually have any awkward features; we therefore assume that they can be described mathematically as volume regions and smooth surfaces which satisfy the conditions set out in Chapter 5. We also assume that any curves in physical space can be validly represented as piecewise smooth curves. Volumes, areas of surfaces and lengths of curves can therefore be defined. We can also define integrals of functions along curves, over surfaces and throughout volumes, provided that the region is finite and the functions are piecewise continuous; these definitions are based on analytic theorems established in Appendix A.

In this chapter, we shall be studying the analysis of functions in 3-space which might represent the properties of, for example, fluids, gravitational or electromagnetic fields, stress and strain in solids, or wave functions in atoms, molecules and nuclei. Such a function may be unbounded when the position vector tends to certain points, curves or surfaces; for example the electrical potential of a point charge e at the origin is e/r, which is unbounded near the origin (r = 0). Special care must be taken in the study of functions in regions where they are unbounded; we shall concentrate our attention on the analysis of functions in regions where they are ‘well-behaved’. By ‘well-behaved’ we not only mean that a function is continuous, but also that any derivatives we use exist and are continuous.