The methods described in this chapter for the solution of linear equations spring from that taught to school children, elimination of the variables until a single equation in a single unknown is reached. In their hands, the method often leads to the conclusion that 0 = 0. To avoid this conclusion being drawn when it does not reflect any special property of the equations, the process will be systematised. The number of arithmetic operations (essentially multiplications) required to solve a set of equations of order n varies as n3. For quite moderate values of n, say 20, this provides plenty of opportunity for mistakes. A running check technique will be described to detect faults before much work based on wrong values has been carried out.

The large number of operations involved has a further consequence which is still of importance when automatic digital computers are used to carry out the arithmetic and checking techniques. This is that the number of round-off errors that are made can affect the result of a calculation to a very great extent. When work is carried out by hand, a skilled computer will see what is happening, and take avoiding action if it becomes necessary. The machine is not endowed with any native intelligence, and if the programmer does not provide for any necessary avoiding action, the machine will continue to run.

The analysis of systems having finite freedom is found to require manipulation of linear algebraic equations. These equations are clumsy in form, particularly if the system to which they relate has more than three or four degrees of freedom, so that some form of algebraic shorthand becomes desirable. Matrix notation serves this purpose. The essence of matrix methods is orderliness, and this feature makes the techniques particularly helpful in the setting out of problems for numerical solution by digital computers or by semi-skilled human computers.

The theme of this presentation of vibration theory is that sets of linear equations may be written concisely in matrix form, by the use of certain conventions. The matrices can then be manipulated (for instance to give other matrices) according to certain rules, re-interpretation at any stage being possible through the conventions. The purpose of this first chapter is merely to provide a background in matrix algebra before the treatment of the dynamical problems is presented.

Matrix notation and preliminary definitions

A matrix is an array of numbers (or ‘elements’), the positions as well as the magnitudes of which have significance. It is thus like a determinant, but with one essential difference.

The theory of vibration which is presented in the previous chapters relates to an idealised vibrating system having a finite number of degrees of freedom. This chapter is concerned mainly with continuous systems, that is, systems with an infinity of degrees of freedom. First it will be shown how receptances may be defined for continuous systems, then it will be shown how composite systems, whether continuous or discrete, may be analysed by the use of matrices. Finally, it will be shown how the theory developed for systems with a finite number of degrees of freedom may be applied to the vibration of continuous systems.

Throughout this chapter only undamped vibration will be discussed. But the reader will observe that much of the analysis may be generalised to cover damped vibrating systems. Throughout this chapter, therefore, all displacements and forces will be harmonic, and will have the same frequency and phase.

The receptances of continuous systems

The receptances of a discrete system—that is, a system with a finite number of degrees of freedom—were defined in § 2.2. Although they were first introduced as elements of the receptance matrix, their fundamental meaning is such that the receptance αxy gives the displacement at the coordinate x due to a harmonic force at the coordinate y.

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.

For reasons that I have outlined in another book, I always advise designers to state their facts and ideas in drawings and diagrams if they can, for words and figures often involve another abstract step. Possibly I may be wise to accept my own advice and so I have prepared a single diagram (Fig. 12) incorporating my conclusions, and my recommendation is that you should have a good look at it.

‘It looks to me like a map of the London Underground gone mad.’

Well, in a sense, it is a map, for it outlines the basic mental and research routes that a designer can follow when searching for data. In a sense it is an index too, for the numbers in circles are those of the chapter in the book dealing with that particular part of the map.

‘What use do you think all that will be?’

I suggest that the first thing you do, when given the responsibility of finding out by research some missing facts that you need for designing, is to take a long hard look at this diagram. It contains, in principle, all the options open to you.

‘What can I do if I don't know enough about a problem to decide in advance where to put it on the map?’

You must do some preliminary research into what is going on. It is surprising how, sometimes at least, you can play about with bits and pieces of things and so quite quickly get the general feel of what it is all about.

We camp with Newton on a special ledge sliced out of the Matterhorn of reality and our experience of this rugged and unpredictable world around us can be interpreted by this fact. As we have seen, we are specially equipped by our senses to live on this unique level. But can we define the basis from which this uniqueness springs?

The answer lies, I suggest, in the fact that our scale of experience lies within that no man's land of physics where relativity and quantum physics intermingle. And the characteristic of this no man's land, where all men live, is that it is under the dictatorship of time. In other spheres time is often a partner, even a junior partner, but with us he is the boss. We cannot opt out from his edicts, our most homely phrases reflect his dictatorship. We talk of ‘half-time’ and ‘overtime’ and alarm clocks and calendars are unlikely to become extinct. None of us can, as yet, escape from that rugged ruler, the ancient with the scythe.

It is the time element that makes life so complicated, and in particular, often makes our research complicated too. We cannot avoid its domination ourselves, but can we filter it out from our experiments? Yes, we sometimes can and, when we can, we must, for the savings are notable.

There are two possible escape routes from time's imprisoning complexities; we must organize our variables so that they either have no relationship with time at all or else have an identical relationship with it.

Longwindedly, but more accurately, this book could have been entitled ‘The General Principles of Scientific Research for Obtaining Data for Engineering Design’.

In the contemporary world of engineering, the designer tends to become a specialist while needing to become a general practitioner. Knowing more and more about less and less, he should be learning more and more about more and more.

This is especially true when he finds that to complete a design he lacks some essential data that can only be found by research.

Now being a good designer does not automatically mean that he is gifted at doing research. Nor does it automatically mean that he is not gifted at it. Neither he, nor the Company or University that employs him, know the answer, and much water may flow under the bridge and much money flow out of the bank before they both can be sure what his aptitude is.

The intention of this book is to ensure that whatever the state of the reader is when he begins reading it, he will, at least, be slightly suitable for carrying out research responsibility by the time he reaches the end of it.

For the old hands I hope a realistically illustrated philosophy based on over forty years experience of research may prove a pattern that will add significance and order to their own.

Finally, this book is one in a series of three, which touch without overlapping, and are intended to cover the whole spectrum of engineering design.

To be good at research you must be good at mountaineering. Scrambling up and down the sloping sides of the problem, you must find the highest level at which its essential features can still be preserved.

Imagine, for instance, that you have been unable to go and watch a special football match but you want to find out what happened. You rush home and begin doing research into the problem by pounding the buttons on your television set until you can find a film recording of the match. You then settle down to watch what happened. And you see what happened by not seeing what happened, not exactly that is.

Your television set only gives you a two-dimensional reproduction of the match; it has dropped a dimension. Now dropping this dimension has not altered the match: the same side wins, but it has enabled you to do the research you wanted on the apparatus you had available. It is special but not misleading. And this dimension-dropping gambit is perhaps the most obvious way to simplify research, provided we do not forget to put it back when it is needed.

For instance, if you are designing a space vehicle, you must study the influence that other distant space objects may have on its flight. For the sake of simplification astronomers assume in calculating these gravitational forces that the total mass of the star or planet is concentrated within a point of no magnitude at its centre.

You can often save money by doing your research on a model of the real problem, though this is not necessarily as simple as it sounds. In making a scaled-up or a scaled-down model you cannot always assume that every dimension can be blindly modified in the same fixed ratio.

Suppose you are still watching your television football match and are shown a distant view to enable you to see the whole field and the progress of the play. The players look like dwarfs. But how can you be certain that they aren't dwarfs? Perhaps the manager of the match has decided to save capital outlay by reducing the size of the pitch and has trained a number of dwarfs to play the game; as they might drink less he wouldn't have to pay them so much. How do you know, merely from watching your television screen, that you are not seeing a consistently scaled-down version of the match? You will probably be able to guess by looking at how the players behave. Soon after the beginning of the match they may well become more interested in kicking each other than in kicking the ball. Why doesn't the referee blow his whistle and send someone off? But the wretched little man is blowing into his whistle with all his might, to no effect. A scaled-down whistle will either produce an unrecognizable squeak or else a sound above the human range of detectable frequencies.

You are still watching this football match on the television and, naturally, it is a scaled down view that you see. But you don't. If you really had a scaled down version of the match in your sitting room, you would see a number of over-heated little dwarfs running about on the carpet in imminent danger of being eaten up by your dog.

You are not seeing a transposition of the game into your room, but the t.v. has provided you with the game faithfully transferred into another medium, where it re-appears as a small point of light dashing around on a special sheet of glass.

The actions of the players are in fact transformed twice. First into radio waves, then into an optical display; and because we are dealing in absolutes their reality is perfectly preserved. We must next consider how this principle may help in expanding the usefulness of model making.

In the last chapter we assumed that our model goals were constructed of the same material as the full size ones, but this is not mandatory. We can, and often do, transform our model by using another medium of construction; some special material that may help us to save time or money.

For instance, our totally transposed goal models are very stiff and their deflection measurements will require highly precise instruments. Moreover the imposed displacements for influence lines technique are likely to involve considerable forces which again are often inconvenient.

In regard to instruments, natural scientists and mechanical scientists have a common interest but a contrasting attitude.

To the natural scientist bigger, better and often more expensive instruments are stepping stones to a triumph, while to the engineer they are more likely to be a short cut to bankruptcy. A natural scientist is often chosen for his ability in evolving instruments as much as for that of being able to interpret what they tell him.

An engineer, if he is wise, will avoid inventing new instruments like the plague (unless that is his actual job) and for three very good reasons.

The first is that he will, almost certainly, be bad at doing it. It is a highly skilled specialized profession where amateurs are likely to make fools of themselves.

Secondly, even if he designs and makes an original one he cannot be sure of its limits of accuracy without a long, probably very long, series of tests. It will be an expensive red-herring.

Thirdly, he must, for the sake both of his own sanity and his sponsors' money pick other people's brains, experience, and production lines as much as possible. He cannot be a one-man band and it is stupid to try.

Don't make instruments, buy them. If you cannot find exactly what you want, modify your research set-up until you can. Only at the very very last resort make your own, and even then do it by a combination or modification of existing ones.