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Dynamics of structures j Jagmohan L. Structural dynamics. Title T A H79 I Typesetting: Macmillan India Ltd.

Printed by: Krips, Meppel, The Netherlands. Ill 3. I 0 Hamil ton's equation I9 Computing speed of FFT convolution. I Introduction. IO Damping orthogonality. I Jacobi diagonalization. I Eigenvalue problem and the orthogonality relationship. I Characteristics of an unrestrained system. Axial vibrations of a bar Torsional vibrations of a bar Transverse vibrations of a string Transverse vibrations of a shear beam Transverse vibrations of a beam excited by support motion Effect of axial force on transverse vibrations of a beam Frequencies and mode shapes for the axial vibrations of a bar Frequencies and mode shapes for the transverse vibration of a string Boundary conditions containing the eigenvalue Free-vibration response of a continuous system Undamped free transverse vibrations of a beam Damped free transverse vibrations of a beam The book should be equally helpful to persons working in the field of civil, mechanical or aerospace engineering.

For the proper understanding of an analytical concept, it is useful to develop an appreciation of the mathematical basis.

Such appreciation need not depend on a rigorous treatment of the subject matter; a physical understanding of the concepts is in most cases adequate, and perhaps more meaningful to an engineer. The book attempts to explain the mathematical basis for the concepts presented, mostly in physically motivated terms or through heuristic argument. No special mathematical background is required of the reader, except for a basic knowledge of college algebra and calculus and engineering mechanics.

The essential steps in the dynamic analysis of a system are: a mathematical modeling b formulation of the equations of motion, and c solution of the equations. Modeling techniques can be divided into two broad categories. In one technique, the system is modeled as an assembly of rigid body masses and massless deformable elements. Systems modeled in this manner are referred to as discrete parameter systems.

In the other technique of modeling, both mass and deformabilty are assumed to be distributed throughout the extent of the system which is treated as continuous. Systems modeled in this manner are called continuous or distributed parameter systems. In general, a continuous model will better represent the behavior of a dy- namical system. However, in most practical situations, the equations of motion of a continuous system are too difficult or impossible to solve.

Therefore, in a majority of cases, dynamic analysis of engineering structures must rely on a representation of the structure by a discrete parameter model. The contents of the book reflect this emphasis on the use of discrete models. The first three parts of the book are devoted to the analysis of response of discrete systems. Part 1, consisting of Chapters 2 through 4, deals with the formulation of equa- tions of motion of discrete parameter systems.

However, the methods of ana- lytical mechanics presented in Chapter 4 are, equally applicable to continuous systems. Examples of such applications are presented later in the book. XVI Humar Part 2 of the book, covering Chapters 5 through 9, deals with the solution of equation of motion for a single-degree-of-freedom system.

Part 3, consisting of Chapters I 0 through 13, discusses the solution of equations of motion for multi degree-of-freedom systems. Part 4 of the book, covering Chapters 14 through 17, is devoted to the analysis of continuous system. Again, the subject matter is organized so that the formulation of equations of motion is presented first followed by a discussion of the solution techniques.

The book is organized so as to follow the logical succession of steps in- volved in the analysis. Many readers may prefer to complete a study of the single-degree-of-freedom systems, from formulation of equation to their solu- tion, before embarking on a study of multi-degree-of-freedom systems.

This can be easily achieved by selective reading. The book chapters have been planned so that Chapters 3 and 4 relating to the formulation of equations of motion of a general system need not be studied prior to studying the material in Chapters 5 through 9 on the solution of equations of motion for a single-degree-of-freedom system. A development that has had a profound effect in the recent times on proce- dures for the analysis of engineering systems is the advent of digital computers.

The ability of computers to manage vast amounts of information and the incred- ible speed with which they can process numerical data has shifted the emphasis from closed from solutions and approximate methods suitable for hand compu- tations to solution of discrete models and numerical techniques of analysis.

At the same time, computers have allowed the routine solution of problems vastly greater in size and complexity than was possible only a decade or two ago. The emphasis on discrete methods and numerical solutions is reflected in the contents of the present book. Chapter 8 on single-degree-of-freedom systems and Chapter 13 on multi-degree-of-freedom systems, are devoted exclusively to numerical techniques of solution. A fairly detailed treatment of the frequency domain analysis is included in Chapters 9 and 13, in recognition of the effi- ciency of this technique in the numeric computation of response.

Also, a detailed treatment of the solution of discrete eigenproblems which plays a central role in the numerical analysis of response is included in Chapter It is recognized that the field of computer hardware as well as software is undergoing revolutionary development. The continuing evolution of personal computers with vastly improved processing speeds and memory capacity and the ongoing development of new programming languages and software tools means that algorithms and programming styles must continue to change to take advantage of the progress made.

Program listings or detailed algorithms have not therefore been included in the book. The author believes that in a book like this, it is more useful to provide the necessary background material for an appreciation of the physical behavior and the analytical concepts involved as well as to present the development of methods that are suitable to numeric Preface XVII computations.

The material included in this book has been drawn from the vast wealth of available information. Some of it has now become a part of the historical development of structural dynamics, other is more recent. It is difficult to ac- knowledge the sources for all of the information provided.

The author offers his apologies to all researchers who have not been adequately recognized. Refer- ences have been omitted from the text to avoid distracting the reader. However, where appropriate, a brief list of suitable material for further reading is provided at the end of each chapter. The style of presentation and the emphasis are the author's own. The contents of the book have been influenced by the author's experience in teaching and research and by the research studies carried out by him and his students.

A large number of examples have been included in the text; since they provide the most effective means of developing an understanding of the concepts involved.

Exercise problems have also been included at the end of each chapter. They will provide the reader useful practice in the application of techniques presented. In preparing this second edition, the errors that had inadvertently crept into the first edition have been corrected. The author is indebted to all those readers who brought such errors to his attention. Several sections of the book have been revised and some new concepts and analytical techniques have been included to make the book as comprehensive as possible, within the boundary of its scope.

Also included are additional end-of-chapter exercises for the benefit of the reader. The author wishes to acknowledge the contribution made by his many stu- dents and colleagues in the preparation of this book. List of symbols The principal symbols used in the text are listed below. All symbols, includ- ing those listed here, are defined at appropriate places within the text, usually at the time of their first occurrence.

Occasionally, the same symbol may be used to represent more than one parameter, but the meaning should be quite unambiguous when read in context. Throughout the text, matrices are represented by bold face upper case letters while vectors are represented by bold face lower case letters.

An overdot signi- fies differential with respect to time and a prime stands for differentiation with respect to the argument of the function. The physical object whose response is sought may either be treated as rigid-body or considered to be deformable.

The subject of rigid-body dynamics treats the physical objects as rigid bodies that undergo motion without deformation when subjected to dy- namic loading. The study of rigid-body motion has many applications, including, for example, the movement of machinery, the flight of an aircraft or a space vehicle, and the motion of earth and the planets.

In many instances, however, dynamic response involving deformations, rather than simple rigid-body motion, is of primary concern. This is particularly so in the design of structures and structural frames that support manufactured objects. Structural frames form a part of a wide variety of physical objects created by human beings: for exam- ple, automobiles, ships, aircraft, space vehicles, offshore platforms, buildings, and bridges. All of these objects, and hence the structure supporting them, are subjected to dynamic disturbances during their service life.

Dynamic response involving deformations is usually oscillatory in nature, in which the structure vibrates about a configuration of stable equilibrium. Such equilibrium configuration may be static, that is, time invariant, or it may be dynamic involving rigid-body motion. Consider, for example, the vibrations of a building under the action of wind. In the absence of wind, the building structure is in a state of static equilibrium under the loads acting on it, such as those due to gravity, earth pressure, and so on.

When subjected to wind, the structure oscillates about the position of static equilibrium as shown in Figure 1. An airplane in flight provides an example of oscillatory motion about an equilibrium configuration that involves rigid-body motion. The aircraft can be idealized as consisting of rigid-body masses of fuselage and the engines con- nected by flexible wing structure Fig.

When in flight, the whole system moves as a rigid body and may, in addition, be subjected to oscillatory motion transverse to the flight plane.

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