Lecture Notes in En~ineering The Springer-Verlag Lecture Notes provide rapid (approximately six months), refereed publication of topical items, longer than ordinary journal articles but shorter and less formal than most monographs and textbooks. They are published in an attractive yet economical forma~ authors or editors provide manuscripts typed to specifications, ready for photo-reproduction. The Editorial Board Managing Editors C. A Brebbia S.A Orszag Dept. of Civil Engineering Dept. of Applied Mathematics University of Southampton Rm 2-347, MIT Southampton S09 5NH (UK) Cambridge, MA 02139 (USA) Consulting Editors Materials Science and Computer Simulation: S. Yip Chemical Engineering: Dept. of Nuclear Engg., MIT J. H. Seinfeld Cambridge, MA 02139 (USA) Dept. of Chemical Engg., Spaulding Bldg. Calif. Inst. of Technology Mechanics of Materials: Pasadena, CA 91125 (USA) F. A Leckie College of Engineering Dynamics and Vibrations: Dept. of Mechanical and Industrial Engineering P. Spanos Univ. of Illinois at Urbana-Ghampaign Department of Mechanical and Civil Engineering Urbana, IL 61801 (USA) Rice University A R. S. Ponter P. O. Box 1892 Dept. of Engineering, The University Houston, Texas 77251 (USA) Leicester LE1 7RH (UK) Earthquake Engineering: Fluid Mechanics: AS. Cakmak K.-P' Holz Dept. of Civil Engineering, Princeton University Inst. fUr Stromungsmechanik, Princeton, NJ 08544 (USA) Universitat Hannover, Callinstr. 32 D-3000 Hannover 1 (FRG) Electrical Engineering: P. Silvester Nonlinear Mechanics: Dept. of Electrical Engg., McGill University K.-J. Bathe 3480 University Street Dept. of Mechanical Engg., MIT Montreal, PO H3A 2A7 (Canada) Cambridge, MA 02139 (USA) Geotechnical Engineering and Geomechanics: Structural Engineering: C.S. Desai J. Connor College of Engineering Dept. of Civil Engineering, MIT Dept. of Civil Engg. and Engg. Mechanics Cambridge, MA 02139 (USA) The University of Arizona W. Wunderlich Tucson, AZ 85721 (USA) Inst. fUr Konstruktiven Ingenieurbau Ruhr-Universitat Bochum Hydrology: U niversitatsstr. 150, G.Pinder D-4639 Bochum-Ouerenburg (FRG) School of Engineering, Dept. of Civil Engg. Prinecton University Structural Engineering, Fluids and Princeton, NJ 08544 (USA) Thermodynamics: J. Argyris Laser Fusion - Plasma: Inst. fUr Statik und Dynamik der R. McCrory Luft-und Raumfahrtkonstruktion Lab. for Laser Energetics, University of Rochester Pfaffenwaldring 27 Rochester, NY 14627 (USA) D-7000 Stuttgart 80 (FRG) Lecture Notes in Engineering Edited by C. A. Brebbia and S. A. Orszag 10 c. A. Brebbia, H.Tottenham, G. B. Warburton, J. M. Wilson, R. R.Wilson Vibrations of Engineering Structures Spri nger-Verlag Berlin Heidelberg New York Tokyo Series Editors C. A Brebbia . S. A Orszag Consulting Editors J. Argyris . K.-J. Bathe· A S. Connor· J. Connor· R McCrory C. S. Desai· K.-P. Holz . F. A Leckie· L. G. Pinder· A R S. Pont J. H. Seinfeld . P. Silvester· P. Spanos· W. Wunderlich· S. Yip Authors C.ABrebbia H.Tottenham Civil Engineering Dept. 11 Middlebridge Street University of Southampton Romsey, Hampshire Southampton S058HJ U.K. U.K. G. B. Warburton· J.M.Wilson Dept. of Mechanical Engg. Dept. of Engineering The University of Nottingham University of Durham University Park Science Laboratories Nottingham NG 7 2 RD South Road U.K. Durham DH 1 3 LE U.K. RR Wilson James Howden &C ompany Limited 195, Scotland Street Glasgow, G 5 8 PJ Scotland ISBN-13: 978-3-540-13959-1 e-ISBN-13: 978-3-642-82390-9 001: 10.1007/978-3-642-82390-9 Library of Congress Cataloging in Publication Data Main entry under title: Vibrations of engineering structures. (Lecture notes in engineering; 10) 1. Structural dynamics. 2. Vibration. I. Brebbia, C. A. II. Series. TA654.V49 1985 624.1'76 84-24800 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © Springer-Verlag Berlin, Heidelberg 1985 2061/3020-543210 FOREWORD The increasing size and complexity of new structural forces in engineering have made it necessary for designers to be aware of their dynamic behaviour. Dynamics is a subject which has traditionally been poorly taught in most engineering courses. This book was conceived as a way of providing engineers with a deeper knowledge of dynamic analysis and of indicating to them how some of the new vibrations problems can be solved. The authors start from basic principles to end up with the latest random vibration applications. The book originated 1n a week course given annually by the authors at the Computational Mechanics Centre, Ashurst Lodge, Southampton, England. Special care was taken to ensure continuity in the text and notations. Southampton 1984 CONTENTS Page Foreword Chapter 1 Introduction to Vibration 1. Introductory Remarks 1 2. Single Degree of Freedom Systems: Equations of Motion and Types of Problem 2 3. Response 6 4. General Structures: Equations of Motion 11 5. Response 15 6. Dynamic Interaction Problems 20 Chapter 2 Free Vibration, Resonance and Damping l. Introduction 25 2. Spring-Mass System 25 3. Simple Pendulum 27 4. Beam with Central Load 28 5. Rolling of a Ship 28 6. Springs in Parallel 30 7. Springs in Series 30 8. Free Vibration 31 9. Energy of Vibrating System 33 10. Damped Free Vibration 34 11. Undamped Forced Response 38 12. Damped Forced Response 39 13. Undamped Transient Vibration 42 14. Damped Transient Vibration 43 15. Summary of Results 44 Chapter 3 Vibrations of Multi-Degree of Freedom Systems 1. Introduction 45 2. Free Vibrations of Two Degree of Freedom Systems 46 3. Free Vibrations of a Multi-Degree of Freedom System 50 4. Orthogonality of Mode Shapes 53 5. Modal Decomposition 55 6. Damped Free Vibrations of Multi-Degree of Freedom Systems 60 7. Forced Vibrations of Multi-Degree of Freedom Systems 62 Chapter 4 Eigenvalue-Eigenvector Solution 1. Introduction 64 2. Three Degree of Freedom System 64 3. Zeros of Determinants 71 4. Banded and Symmetric Matrices 71 5. Reduction of Eigenvalue Equation to Standard Form 72 6. Solution of Standard Eigenvalue Equations by Sturm Sequence Technique 74 7. Solution of the Original Equations using Sturm Sequence Technique 75 8. Simultaneous Iteration 76 9. Comparison of Eigenvalue Solution Methods 78 10. Node Condensation 80 11. Substructure Analysis 81 12. Rate of Change of Eigenvalues 82 Chapter 5 Approximate Methods for Calculating Natural Frequencies and Dynamic Response of Elastic Systems 1. Equivalent One Degree of Freedom Systems 84 2. Continuous Beams 85 3. Distribution Methods 88 4. Multi-Storey Frames 90 Chapter 6 Determination of Response 1. Introductory Remarks 92 2. Steady State Response 93 3. Damping 95 4. Truncation of Series Solution 97 5. Response Spectrum Methods 101 "Chapter 7 The Finite Element Technique 1. Introduction 104 2. The Principle of Virtual Displacements 106 3. Finite Element Discretization and Element Matrices no 4. System Equations 122 5. Solution 126 Chapter 8 Two Dimensional and Plate Bending Applications 1. Introduction 130 2. In-Plane Plate Elements 132 3. In-Plane Vibration of Plates 136 4. Plate Bending Elements 137 5. Transverse Vibration of Plates 140 6. Combination of Plate and Beam Elements 143 Chapter 9 Transient Response of Structures 1. Introduction 151 2. Transient Response without Damping 151 3. Damping 154 4. Damped Transient Response 160 5. Numerical Methods 165 Chapter 10 Machine Foundations 1. Introduction 176 2. Transmissibility of a Foundation on a Rigid Base 176 3. Transmissibility of a Foundation on a Flexible Base 180 4. Low Tuned and High Tuned Foundations 184 5. Dynamic Absorber 186 6. Damped Dynamic Absorber 188 7. Design Codes 190 8. Steel Foundations for Turbo-Alternators 191 9. Conclusions 193 VI Chapter 11 Vibration ofAxi-Symmetric Shells 1. Introduction 195 2. Novozhilov's Thin Shell Theory 195 3. Finite Element Displacement Method applied to Axi-Symmetric Shells 203 4. Vibration Applications 208 5. Example 209 Appendix. Matrices used in the Text 211 Chapter 12 Some Recent Advances in Structural Vibration l. Introductory Remarks 215 2. Direct Integration Methods 217 3. Accuracy 219 4. Non-linear Problems 221 5. Partitioning 222 Chapter 13 Fluid Structure Interaction Problems 1. Introduction 225 2. The Mechanics of Drag, Inertia and Lift 228 3. Total Hydrodynamic Forces 248 4. Final Remarks 248 Chapter 14 Introduction to Random Vibrations 1. Random Processes 251 2. Spectral Density Function 255 3. The Weiner-Khinchin Relationship 258 4. Response of a Single Spring System to Random Load 260 Chapter 15 Earthquake Response of Structures 1. Introduction 265 2. Beam Analysis 266 3. Spectral Density of Response 268 Chapter 16 Response of Structures to Wind Loading 1. Introduction 273 2. Response of Shells 276 Chapter 17 Random Response Analysis of Off-Shore Structures 1. Introduction 280 2. One Degree of Freedom System 282 3. Multi-Degree of Freedom System 288 4. Closing Remarks 299 CHAPTER 1 INTRODUCTION TO VIBRATION by G.B. Warburton 1. Introductory Remarks In recent years the number of structures, for which the dynamic forces, likely to be encountered in service, have required investigation at the design stage, has increased. Several factors have contributed to this increase: growth in size of structures of various types; consequential increased importance of wind forces; efforts to reduce the effects of earthquakes on structures and to prevent total collapse; design of off-shore structures. Two important questions are: why is it essential to include dynamic effects in structural analysis and why is this a more difficult task than conventional (static) structural analysis? Suppose that the stresses in a structure are known for: (a) a static force P at a particular location; (b) a force at the same location that varies in magnitude with time and has a maximum value of P. Then the dynamic magnification factor is the maximum stress at a point for (b) / the stress at the same point for (a). This factor depends upon how the force varies with time, the distribution of stiffness and mass in the structure and the damping present. In certain circumstances it will be very large; in others very small. Obviously, if there is any possibility of the dynamic magnification factor being significantly greater than unity, a dynamic analysis of the structure is necessary. This book is primarily concerned with methods of determining dynamic magnification factors for various types of loads and structures. However, no simple rules exist for these factors. Thus there are greater conceptual difficulties for dynamic problems than for comparable static problems, as the intuition and experience, which help an engineer to form a reason able view of the safety of a structure under static forces, do not lead to an esti mate of the relevant dynamic magnification factors. Also the time dependence of stresses, displacements etc. and the necessity to include mass and damping effects make dynamic analysis more complex than its static counterpart. There are also practical difficulties; some dynamic loads, e.g. wind forces, and most damping forces can only be estimated. In addition to the possibility of elastic failure of a structure if dynamic effects are neglected, long-time repetition of dynamic stresses, whose magnitudes would be considered to be safe from static considerations, may lead to cumulative fatigue failures. 2 In this chapter the concepts that are relevant to vibration analysis of structures will be discussed briefly. Emphasis is on the response of structures to dynamic forces and how different types of force time variation influence the choice of method. Many of the concepts are introduced by considering the simplest vibrating structure; then, as this simple structure has limited practical applications, gen eral structures are discussed. For these the normal mode method of determining response is given particular attention, because it illustrates the physical behaviour of structures better than other methods. Lastly dynamic interaction problems are discussed; here interaction exists between the vibrations of a structure and those of the underlying soil or the surrounding fluid. Many current practical problems, and also much current research effort, involve interaction effects. Naturally in a single chapter the major topics of structural vibration can only be mentioned. Most of these topics will be studied in depth in subsequent chapters. It is hoped that their introduction here will illustrate their interrelationship and show how they contribute to the determination of stresses in complex structures caused by various types of dynamic excitation. 2. Single Degree of Freedom Systems: Equation of Motion and Types of Problem Although the dynamic response of a practical structure will be complex, it is necess ary to begin our study by considering the fundamentals of vibration of simple systems. A rough guide to the complexity of a dynamical system is the number of degrees of freedom possessed by the system. This number is equal to the number of independent coordinates required to specify completely the displacement of the system. For instance, a rigid body constrained to move in the X Y plane requires three coord inates to specify its position completely - namely the linear displacements in the X- and Y-directions and the angular rotation about the Z-axis (perpendicular to the plane X Yl; thus this body has three degrees of freedom. The displacement of an elastic body, e.g. a beam, has to be specified at each point by using a continuous equation so that an elastic body has an infinite number of degrees of freedom. In a dynamical problem the number of modes of vibration in which a structure can respond is equal to the number of degrees of freedom, thus the simplest structure has only one degree of freedom. Figure 1 shows the conventional representation of a system with one degree of free dom; it consists of a mass m constrained to move in the X-direction by frictionless guides and restrained by the spring of stiffness k. It is assumed that the mass of the spring is negligible compared to m. Thus the displacement of the system is specified completely by x, the displacement of the mass, and the system has one degree of freedom. For the purpose of analysing their dynamic response it is possible to treat some simple structures as systems with one degree of freedom. 3 A Figure I Single degree of freedom system --I x,l -- -j, x,l - I pet) I B C \ \ \ \ \ \ \ A D Figure 2 Simple frame with one degree of freedom pet) pet) P P c 0 o t t (a) (b) Figure 3 Examples of transient force excitation