There are many books on finite element methods but very few give more than a brief description of their application to structural vibration analysis. I have given lecture courses on this topic to undergraduates, postgraduates and those seeking post experience training for many years. Being unable to recommend a single suitable text led me to write this book.
The book assumes no previous knowledge of finite element methods. However, those with a knowledge of static finite element analysis will find a very large proportion of the book useful. It is written in such a way that it can be used by Aeronautical, Civil, Mechanical and Structural Engineers as well as Naval Architects. References are given to applications in these fields.
The text has been written in modular style. This will facilitate its use for courses of varying length and level. A prior knowledge of strength of materials and fundamentals of vibration is assumed. Mathematically, there is a need to be able to differentiate and integrate polynomials and trigonometric functions. Algebraic manipulation is used extensively but only an elementary knowledge of vector methods is required. A knowledge of matrix analysis is essential. The reader should be able to add, subtract, multiply, transpose, differentiate and integrate matrices. Methods of solving linear equations and the existence of a matrix inverse is a prerequisite and the evaluation of determinants is also required.
Chapter 1 deals with methods of formulating the equations of motion of a dynamical system. A number of methods are introduced. The advantages and disadvantages of each one are discussed and recommendations made. The treatment is simple for ease of understanding with more advanced aspects being treated in an Appendix. The simplest methods derive the equations of motion from the expressions for kinetic and strain energy and the virtual work done by externally applied loads. Expressions for these are derived for various structural elements in Chapter 2.
The response of practical structures cannot be obtained using analytical techniques due to their complexity. This difficulty is overcome by seeking approximate solutions. Chapter 3 begins by describing the technique known as the Rayleigh-Ritz method. The finite element displacement method is
then introduced as a generalised Rayleigh-Ritz method. The principal features of the method are introduced by considering rods, shafts, beams and frameworks. In this chapter specific element matrices are evaluated explicitly. However, many of the elements presented in later chapters can only be evaluated using numerical integration techniques. In preparation for this, numerical integration in one dimension is introduced. The extension to two and three dimensions is presented where required.
In Chapter 4 various membrane elements are derived. These can be used for analysing flat plate structures which vibrate in their plane. Chapter 5 deals with the vibration of solids using both axisymmetric and three- dimensional elements. Chapter 6 indicates the difficulties encountered in the development of accurate plate-bending elements. This has led to a large number of elements being developed in attempting to overcome these problems. Chapter 7 describes methods of analysing the vibrations of stiffened plates and folded plate structures. This involves combining the framework, membrane and plate bending elements described in previous chapters. The problems which arise and how to overcome them are described.
Chapters 8, 9 and 10 present methods of solving the equations of motion. Chapter 8 considers the equations for free vibration of an undamped structure. These take the form of a linear eigenproblem. The methods of solution to be found in the major finite element systems are described. The presentation is designed to give the finite element user an appreciation of the methods. Program developers will need to consult the references given for further details. Methods of reducing the number of degrees of freedom are presented. These consist of making use of symmetry, the analysis of rotationally period structures, Guyan reduction and component mode syn- thesis.
Methods of predicting the response of structures to harmonic, periodic, transient and random loads are described in Chapters 9 and 10. Both direct and modal analysis techniques are presented. Methods of representing damping are discussed. The prediction of the response to transient loads involves the use of step-by-step integration methods. The stability and accuracy of such methods are discussed. The response to imposed displace- ments and response spectrum methods are also considered.
The final chapter on Computer Analysis Techniques assumes that the reader intends to use a commercial program. Those wishing to write pro- grams are referred to suitable texts.
Chapters 3 to 7 present details of the simpler elements. Reference to more advanced techniques are given at the end of each chapter. Each one has its own extensive list of references. Throughout the book numerical examples are presented to illustrate the accuracy of the methods described.
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