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Unformatted text preview: Risk Engineering Junbo Jia Essentials of Applied Dynamic Analysis Risk Engineering Series editor Dirk Proske, Vienna, Austria For further volumes: The Springer Book Series Risk Engineering can be considered as a starting point, looking from different views at Risks in Science, Engineering and Society. The book series publishes intense and detailed discussions of the various types of risks, causalities and risk assessment procedures. Although the book series is rooted in engineering, it goes beyond the thematic limitation, since decisions related to risks are never based on technical information alone. Therefore issues of ‘‘perceived safety and security’’ or ‘‘risk judgment’’ are compulsory when discussing technical risks, natural hazards, (environmental) health and social risks. One may argue that social risks are not related to technical risks, however it is well known that social risks are the highest risks for humans and are therefore immanent in all risk trade-offs. The book series tries to cover the discussion of all aspects of risks, hereby crossing the borders of scientific areas. Junbo Jia Essentials of Applied Dynamic Analysis 123 Junbo Jia Aker Solutions Bergen Norway ISSN 2195-433X ISBN 978-3-642-37002-1 DOI 10.1007/978-3-642-37003-8 ISSN 2195-4348 (electronic) ISBN 978-3-642-37003-8 (eBook) Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2013955237  Springer-Verlag Berlin Heidelberg 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media ( ) To Jing and Danning, who make life a gift v Preface The subject of dynamics originated from Sir Isaac Newton’s monograph Philosophiæ Naturalis Principia Mathematica, and Lord Rayleigh paved the way for its further development with his Theory of Sound. These provided the basis for the unique position of the field of dynamics in mechanics. Since then, many scientists and engineers have applied and furthered this knowledge in various fields of applied science and technology. With the enormous investment made in civil, mechanical, and aerospace engineering during the twentieth century, designs were pushed to the limits of their performance capacity, with the trend being toward high-speed operations, adverse environment capability, light weight, etc. With the requirement of functionality in an unpredictable, highly uncertain environment, practitioner engineers encountered more and more problems with regard to dynamics. Pure mathematics is sometimes satisfied with showing that the non-existence of a solution implies a logical contradiction, while engineers might consider numerical results the desirable goal. Although dynamics as a scientific topic is by no means fully understood (and perhaps never will be), the great amount of activity in this field during the last century has made it possible to form a practical subject in a fairly systematic, coherent, and quantitative manner. All these factors have pushed applied dynamics into a greater complexity than it has ever had before, and also promoted the subject into one of the essential tools in current engineering. Thanks to the rapid development of computer technology, more portable and accurate testing equipment and techniques, as well as a few breakthroughs in computation algorithms, during the last 50 years applied dynamics has found efficient and unique ways of developing. This raised a vast amount of challenges in implementing designs in reality, while also putting ever higher demands on engineers, requiring a thorough understanding of the subject. In spite of increased engineering knowledge, the practical problems regarding dynamics and vibrations are in some cases handled without success despite large expenditures of money. Moreover, even if engineers can perform sophisticated computer-based dynamic analysis tasks, many of them lack an actual understanding of the essential principles of dynamics, and hence of the links between theory and application. This leads to an insurmountable barrier when they are requested to validate/verify and provide insightful explanations of analysis results, or to further improve the engineering designs with regard to vibrations, which poses a significant safety vii viii Preface hazard and can also result in significant economic loss. These considerations motivated the author to write this book. With the objective of providing up-to-date knowledge of dynamic analysis, which is of great importance from the point of view of engineering, in the preparation of the book, the author tried to link the general principles of dynamics with their applications from various angles in order to make it possible for readers from various backgrounds to appreciate their significance. The book aims to be as elegant as is possible given this wide-ranging treatment of the subject. The book is intended to serve as an introduction to the subject and also as a reference book with advanced topics. A balance between the theoretical and practical aspects is sought. All the chapters are addressed to practitioner engineers who are looking for answers to their daily engineering problems, and to students and researchers who are looking for links between theoretical and practical aspects, and between phenomena and analytical explanations. It should also be of use to other science and engineering professionals and students with an interest in general dynamic analysis. The book is written in such a way that it can be followed by anyone with a basic knowledge of structural analysis. The mathematical background assumed for reading this book is a working knowledge of differential equations, matrix manipulation, and an elementary knowledge of statistics/probability. In addition, readers are also assumed to have basic knowledge on the strength of materials. The book covers topics on the concepts, principles, and solutions of dynamics and vibrations. These are essential for engineers and researchers to further explore any type of dynamic analysis, such as mechanical vibrations or dynamic structural responses due to environmental loads such as wave, wind, earthquake, and ice loading, etc. The core knowledge of linear and nonlinear dynamics, damping effects, random vibrations, and modal analysis is elaborated. The various solution schemes and selection criteria for a given problem are discussed. The modeling and measuring of damping are also elaborated. Special topics on seismic responses, fatigue assessment, human body vibrations, and vehicle-structure interactions are discussed. The engineering applications, relevant codes and practice, and their links with theory are also provided in relevant chapters. The first three chapters present and discuss the phenomena, concepts, and principles of dynamic analysis with discussions on their applications. Chapter 1 gives an introduction to dynamics in the physical world, distinguishes its essential differences from its static counterpart, and briefly summarizes general methods for treating a dynamic problem. Chapter 2 elaborates the basic formulation of governing equations of motions, which include the formulation of and relationships among Newton’s second law of motions, Hamilton’s principle, and Lagrange’s equation, the three pillars of classical dynamics. This chapter also provides preparatory work for solving both free (Chap. 3) and forced (Chap. 11) vibration problem. Between chapters on free and forced vibrations, important topics focusing on eigenfrequencies and mode shapes are examined in Chap. 4 (for presenting eigenanalysis for discrete systems and a brief introduction on vibration-based structural health monitoring), Chap. 5 (eigenproblem for continuous system), Chap. 6 Preface ix (vibration under axial load), and Chap. 7 (eigenproblem for nonuniform beams and foundations). Note that explicit and concise equations to describe a dynamic system and its responses, like a deterministic one such as Newton’s, is seldom able to reflect realworld phenomena, which are complex, noisy, high-dimensional, etc., and for which the instantaneous value cannot be explicitly predicted at any time instant or reproduced. These can be treated by statistical description and characterizing randomness (probability distribution) of loads and responses, which promoted the research and application of stochastic dynamics. Therefore, Chaps. 8 and 9 systematically examine the deterministic and stochastic loads and responses from a statistical point of view. The essential concepts of Fourier and power spectrum as well as the relationship between a spectrum and its statistical properties are discussed. These form the pillar for stochastic dynamics, which is in parallel to and promotes a wider application of Newton’s equations. In Chap. 10, concepts of short and long-term probability distribution and number of occurrence are introduced. They pave the way for a reasonable understanding of load level at a given return period and for a further extension to reliability and risk assessment. This is also a part of background knowledge for assessing fatigue damage due to dynamic loading (Chap. 17). With the understanding of spectrum analysis and power spectrum (Chap. 9), the power spectrum densities due to specific environment loads with wind, wave, ice, and earthquake loadings are presented in Chap. 12. As Chaps. 8, 9 and 10 provide a broad overview of loads and responses, they enable efficient solutions for forced vibration problems as elaborated in Chap. 11. When reading Chap. 11, readers need to bear in mind that if the excitations are of a deterministic nature, a direct solving of equations of motions is preferred. However, if excitations are of strong stochastic nature, a random vibration approach is more efficient. In Chap. 13, the solution to the dynamic responses is extended from a singledegrees-of-freedom to a multi-degrees-of-freedom system. In addition, the most popular numerical methods (i.e., the direct/exact method, modal superposition method, and the direct integration methods) are discussed with an emphasis on their applicabilities. As the estimation and modeling of the damping are rather difficult tasks for both engineering and research purposes, and in the meantime the resulting uncertainties with damping pose a great challenge to reach a reasonable accuracy for the calculated dynamic responses (a phenomenon more apparent for dynamic sensitive structures), Chap. 14 is therefore dedicated to an elaboration of the effects, modeling, and measuring of various types of damping. As almost all applied processes exhibit nonlinearities in various forms and extents, it is of particular importance to study nonlinear dynamics and vibrations. Therefore, Chap. 15 elaborates this topic by distinguishing them from their linear counterpart, summarizing their causes and sources, and by presenting the relevant numerical solution strategies used in engineering practice. x Preface For dynamic analysis with any extent of difficulty for a real system or a structure, the numerical challenges generally arise from three aspects: space and time discretization and various types of nonlinearities. In the last 60 years, these have attracted extensive research efforts and become almost matured for engineering applications by finite element analysis (for space discretization), finite difference (Newmark’s type) method (for time discretization), and linear iteration (Newton’s type) method (for solving nonlinearities). These three methods form the cornerstones of current applied dynamic analysis. The finite element method can be studied in many available literatures, and the finite difference and linear iteration methods are elaborated in Chaps. 11 and 15, respectively. After digesting the first 15 chapters, readers should have the capability to find solutions of dynamic responses in their specific fields of applications. In Chaps. 16 to 19, the essential knowledge presented in the first 15 chapters is extended to a few of their application areas, with discussions on seismic responses (Chap. 16), fatigue assessment (Chap. 17), human body vibrations (Chap. 18), and vehicledeck dynamic interactions (Chap. 19). While the book does not seek to promote any specific ‘‘school of thought,’’ it inevitably reflects this author’s ‘‘best practice’’ and ‘‘working habit.’’ This is particularly apparent in the topics selected and level of detail devoted to each of them, their sequences, and the choices of many mathematical treatments and symbol notations, etc. The author hopes that this does not deter the readers from seeking to find their own ‘‘best practice’’ and dive into the vast knowledge basin of modern dynamics, which is extremely enjoyable as readers go deeper and wider. Most of the chapters in this book can be covered in a two-day industry course in a brief manner, a one-week intensive course for either industry or university, or a two-semester course in an elaborated form for graduate students. The first four chapters together with Chaps. 11, 13, and 14 can also form a one-semester undergraduate course on structural dynamics or mechanical vibrations. I am indebted to many individuals and organizations for assistance of various kinds, such as participation in book reviews, technical discussions, research co-operation, contributing illustrations, and copyright clearance. These include: Gunnar Bremer, Håkon Sylta, Tore Holmås, Olav Helset Lien, Zhibin Jia, Rikard Mikalsen, Peng Zheng, Vicky McNiff (Aker Solutions), Wai-Fah Chen (University of Hawaii), Andy Ruina (Cornell University), Wengang Mao, Jonas Ringsberg, and Igor Rychlik (Chalmers University of Technology), Douglas Stock (Digital Structures, Inc. Berkeley), Stefan Herion (Karlsruhe Institute of Technology), Anders Ulfvarson (The Royal Swedish Academy of Engineering Sciences), Alaa Mansour (University of California at Berkeley), Salvador Ivorra Chorro (University of Alicante), Christopher Stubbs and Philip Wilmott (Colebrand International Limited), Weicheng Cui (Shanghai Jiaotong University), Tadashi Shibue (Kinki University), Matthew S. Allen (University of Wisconsin-Madison), Derek A. Skolnik (Kinemetrics, Inc), Lance Manuel (University of Texas at Austin), Rune Elleffsen, Terje Nybø, Tor Inge Fossan, and Odd Jan Andersen (Statoil), Ketil Aas-Jakobsen (Dr. Ing. A. Aas-Jakobsen AS), Preben Terndrup Preface xi Pedersen (Technical University of Denmark), Flemming Jacobsen, Martin J. Sterndorff and Lyngberg Kim (Dong Energy), Jeffrey Wang (North America Wave Spectrum Science and Trade Inc), International Organization for Standardization (ISO), International Society of Offshore and Polar Engineers (ISOPE), Springer, Cambridge University Press, and Elsevier. Moreover, there are numerous others not named to whom I extend my sincere thanks. This book has an extensive list of references reflecting both the historical and recent developments on the subject. I would like to thank all the authors in the references for their contribution to the area. I wish to thank all colleagues at Aker Solutions Bergen, especially to those at Structural and Marine Department for providing a technically and socially inspiring working environment. I would also like to acknowledge the support from Concept and Technology at Aker Solutions MMO, especially from Daniel Cazòn, Nils-Christian Hellevig, and Kristian Risdal. Most importantly, I dedicate this book to those who live with me every day, and who brought me into existence. I conclude this preface with an expression of deep gratitude to them. Contents 1 Introduction . . . . . . . . . . . . . . . . 1.1 Experiencing Dynamics . . . . . 1.2 Utilize Dynamics. . . . . . . . . . 1.3 Dynamics Versus Statics . . . . 1.4 Solving Dynamic Problem . . . 1.5 Pioneers of Dynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 14 19 24 29 2 Governing Equation of Motions . . . . . . . . . . . . . . . . . 2.1 Dynamic Equilibrium. . . . . . . . . . . . . . . . . . . . . . 2.2 Principle of Virtual Displacements . . . . . . . . . . . . 2.3 Hamilton’s Principle Through Lagrange’s Equations 2.4 Momentum Equilibrium . . . . . . . . . . . . . . . . . . . . 2.5 Validity of Classical Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 31 33 34 38 39 3 Free Vibrations for a Single-Degree- of-Freedom (SDOF) System–Translational Oscillations. . . . . . . . . . . . . . . . . . 3.1 Definition of Harmonic Oscillations. . . . . . . . . . . . . . 3.2 Undamped Free Vibrations of a SDOF System . . . . . . 3.3 Damped Free Vibrations of an SDOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 41 42 47 Practical Eigenanalysis and Structural Health Monitoring. . . . . . 4.1 Eigenpairs, Global-, Local- and Rigid-Body Vibrations . . . . . . 4.2 Hand Calculation of Natural Frequency for Systems with Distributed Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Classical Method for Exact Solutions . . . . . . . . . . . . . 4.2.2 Equivalent System Analysis for Approximate Solutions. 4.2.3 Natural Frequency with Distributed Masses: Dunkerley Method for Approximate Solutions . . . . . . . 4.3 Using Symmetry and Anti-Symmetry in Eigenanalysis. . . . . . . 4.4 Vibration-Based Structural Health Monitoring. . . . . . . . . . . . . . . 55 55 . . . 58 58 61 . . . 68 72 74 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii xiv 5 Contents Solving Eigenproblem for Continuous Systems: Rayleigh Energy Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6 Vibration and Buckling Under Axial Loading . . . . . . . . . 6.1 Vibration Versus Buckling . . . . . . . . . . . . . . . . . . . . 6.2 Vibration and Buckling Under Harmonic Axial Loads . 6.3 Eigenvalues Under the Influence of Axial Loads. . . . . . . . . 87 87 88 89 7 Eigenfrequencies of Non-uniform Beams, Shallow and Deep Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Non-uni...
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