Fundamentals of Structural Dynamics

By Zhihui Zhou, Ying Wen, Chenzhi Cai and Qingyuan Zeng

Dynamics of Structural Dynamics explains foundational concepts and principles surrounding the theory of vibrations and gives equations of motion for complex systems. The book presents classical vibration theory in a clear and systematic way, detailing original work on vehicle-bridge interactions and wind effects on bridges. Chapters give an overview of structural vibrations, including how to formulate equations of motion, vibration analysis of a single-degree-of-freedom system, a multi-degree-of-freedom system, and a continuous system, the approximate calculation of natural frequencies and modal shapes, and step-by-step integration methods. Each chapter includes extensive practical examples and problems.

This volume presents the foundational knowledge engineers need to understand and work with structural vibrations, also including the latest contributions of a globally leading research group on vehicle-bridge interactions and wind effects on bridges.

• Explains the foundational concepts needed to understand structural vibrations in high-speed railways
• Gives the latest research from a leading group working on vehicle-bridge interactions and wind effects on bridges
• Lays out routine procedures for generating dynamic property matrices in MATLAB©
• Presents a novel principle and rule to help researchers model time-varying systems
• Offers an efficient solution for readers looking to understand basic concepts and methods in vibration analysis

• Language: English
• Publisher: Elsevier Science
• Release date: Jun 8, 2021
• ISBN: 9780128237052

Zhihui Zhou

Zhihui Zhou is an Associate Professor in the School of Civil Engineering at Central South University in China. His research focuses on train derailment and vibration analysis. He has led several research projects, including on the safe operation of high-speed trains over railway bridges, and the safety of long-span cable-stayed bridge trains. He holds a PhD in Bridge and Tunnel Engineering, and has published over 30 papers, and two monographs.

Book preview

Fundamentals of Structural Dynamics

Zhihui Zhou

Central South University, Changsha, Hunan, China

Ying Wen

Central South University, Changsha, Hunan, China

Chenzhi Cai

Central South University, Changsha, Hunan, China

Qingyuan Zeng

Central South University, Changsha, Hunan, China

Table of Contents

Cover image

Title page

Copyright

About the authors

Preface

Chapter 1. Overview of structural dynamics

Abstract

1.1 Objective of structural dynamic analysis

1.2 Characteristics of structural dynamics

1.3 Classification of vibrations

1.4 Vibration problems in engineering

1.5 Procedures of dynamic response analysis of structures

Problems

References

Chapter 2. Formulation of equations of motion of systems

Abstract

2.1 System constraints

2.2 Representation of system configuration

2.3 Real displacements, possible displacements, and virtual displacements

2.4 Generalized force

2.5 Conservative force and potential energy

2.6 Direct equilibrium method

2.7 Principle of virtual displacements

2.8 Lagrange’s equation

2.9 Hamilton’s principle

2.10 Principle of total potential energy with a stationary value in elastic system dynamics

2.11 The set-in-right-position rule for assembling system matrices and method of computer implementation in Matlab

References

Problems

Chapter 3. Analysis of dynamic response of SDOF systems

Abstract

3.1 Analysis of free vibrations

3.2 Response of SDOF systems to harmonic loads

3.3 Vibration caused by base motion and vibration isolation

3.4 Introduction to damping theory

3.5 Evaluation of viscous-damping ratio

3.6 Response of SDOF systems to periodic loads

3.7 Response of SDOF systems to impulsive loads

3.8 Time-domain analysis of dynamic response to arbitrary dynamic loads

3.9 Frequency-domain analysis of dynamic response to arbitrary dynamic loads

References

Problems

Chapter 4. Analysis of dynamic response of MDOF systems: mode superposition method

Abstract

4.1 Analysis of dynamic properties of multidegree-of-freedom systems

4.2 Coupling characteristics and uncoupling procedure of equations of MDOF systems

4.3 Analysis of free vibration response of undamped systems

4.4 Response of undamped systems to arbitrary dynamic loads

4.5 Response of damped systems to arbitrary dynamic loads

References

Problems

Chapter 5. Analysis of dynamic response of continuous systems: straight beam

Abstract

5.1 Differential equations of motion of undamped straight beam

5.2 Modal expansion of displacement and orthogonality of mode shapes of straight beam

5.3 Free vibration analysis of undamped straight beam

5.4 Forced vibration analysis of undamped straight beam

5.5 Forced vibration analysis of damped straight beam

References

Problems

Chapter 6. Approximate evaluation of natural frequencies and mode shapes

Abstract

6.1 Rayleigh energy method

6.2 Rayleigh–Ritz method

6.3 Matrix iteration method

6.4 Subspace iteration method

6.5 Reduction of degrees of freedom in dynamic analysis

References

Problems

Chapter 7. Step-by-step integration method

Abstract

7.1 Basic idea of step-by-step integration method

7.2 Linear acceleration method

7.3 Wilson-θ method

7.4 Newmark method

7.5 Stability and accuracy of step-by-step integration method

Problems

References

Index

Copyright

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About the authors

Dr. Zhihui Zhou is currently an associate professor at the School of Civil Engineering, Central South University (CSU), in China. He received a PhD in Civil Engineering from CSU in 2007 under the supervision of Prof. Qingyuan Zeng. He was invited to study at the University of Kentucky in 2014. Dr. Zhou’s research interests include train derailment and dynamics of train–bridge (track) systems. He has been the principal investigator of several research grants, including the research project of National Natural Science Foundation of China (a study on the control theory of running safety and comfort for high-speed trains on bridges), a scientific research project of China’s Ministry of Railways (a study on safety of running trains on large span cable-stayed bridges), special and general projects of the Chinese Postdoctoral Science Foundation, and some other scientific research projects. Dr. Zhou has published over 30 journal papers as the first author, and two monographs entitled Lectures on dynamics of structures and Theory and application of train derailment. He won the first prize of the Science and Technology Progress of Hunan Province for his study Theory and application of train derailment in 2006.

Dr. Ying Wen was employed in the School of Civil Engineering, CSU, in China, after obtaining his PhD in 2010, and he was promoted to associate professor in 2012. He became a research associate in the Department of Civil and Structural Engineering, The Hong Kong Polytechnic University in 2011. In 2014 Dr. Wen was invited to visit the Department of Aerospace and Mechanical Engineering, University of Southern California, for a collaborative research on the problem of moving loads on structures. After he returned to CSU in 2015, Dr. Wen was appointed as the vice director of the Key Laboratory of Engineering Structures of Heavy-haul Railway, Ministry of Education. Dr. Wen has interests in fields of various structural dynamics and stability, especially nonlinear mechanics of long-span bridges and their dynamic stability under moving trains. Dr. Wen has published more than 20 journal papers, one of which is listed as the Top 25 Hottest articles published in Finite Elements in Analysis & Design. He has also published three Chinese monographs about statics and dynamics of structures as a coauthor. Dr. Wen has received the awards of the Science and Technology Progress of Hunan Province (2006) and Zhejiang Province (2011).

Dr. Chenzhi Cai received his BS degree in civil engineering and MS degree in road and railway engineering from CSU, in China in 2011 and 2015, respectively. He graduated from The Hong Kong Polytechnic University with a PhD in civil engineering in 2018 and joined the Department of Bridge Engineering as well as the Wind Tunnel Laboratory of CSU as an associate professor later that year. Dr. Cai’s main research interests are the fields of noise and vibration control, train-bridge interaction dynamics, and train-induced ground vibration isolation. He has participated in several research projects funded by the Hong Kong government and has also received research funding from the National Natural Science Foundation of China and Hunan Provincial Natural Science Foundation of China. Dr. Cai has published more than 20 papers in international journals, and some of his work is under consideration for acceptance by the UK CIBSE Guide.

Prof. Qingyuan Zeng is a distinguished scientist on bridge engineering at Central South University, in China. He obtained his BS and MS degrees from the Department of Civil Engineering, Nanchang University and Department of Engineering Mechanics, Tsinghua University, in 1950 and 1956, respectively. He was elected as a member of the Chinese Academy of Engineering in 1999 for his great contributions to local–global interactive buckling behavior of long-span bridge structures, train–bridge interaction dynamics and the basic theory of train derailment. He presented the principle of total potential energy with a stationary value in elastic system dynamics and the set-in-right-position rule for assembling system matrices, which is a significant improvement of the classical theory of structural dynamics and finite element method. Prof. Zeng has an international reputation for his originality in the transverse vibration mechanism and time-varying analysis method of the train–bridge system. He has authored and coauthored more than 100 journal papers, three monographs, and three textbooks. He received numerous awards, including the State Science and Technology Progress Award, Distinguished Achievement Award for Railway Science and Technology from Zhan Tianyou Development Foundation, and Honorary Member Award from the China Railway Society. He has supervised more than 16 MS students and 30 PhD students in the past three decades.

Preface

Zhihui Zhou, Ying Wen, Chenzhi Cai and Qingyuan Zeng

Nowadays, the design of engineering structures, for example, long-span bridges, high-rise buildings, stadiums, airport terminals, and offshore platforms, seeks a large ratio of their load carrying capacity to self-weight to achieve esthetic pleasure and economy. However, the type of these lightweight and flexible structures will lead to a large deformation and excessive vibrations under loading. In addition, these structures may suffer from some extreme excitations, for instance, strong winds, seismic actions, high-impact collisions, and impacts of water wave flow. Therefore, investigation of structural behaviors under dynamic loads is essential in order to achieve a good performance of the structure when satisfying the requirement of designed service. The basic concept of structural dynamics is of great help to engineers in understanding structural vibration and taking appropriate measures.

This book introduces the fundamental concepts and basic principles of the dynamics of structures. Although the book focuses on the linear problem in structural dynamics, solutions for some nonlinear problems have also been briefly introduced. It should be noted that random vibration is beyond the scope of this book and is not included here. The main content of this book includes the overview of structural dynamics, the formulation of equations of motion of systems, the analysis of dynamic response of SDOF systems, the analysis of dynamic response of MDOF and continuous systems, the mode superposition method, the approximate evaluation of natural frequencies and mode shapes, and the step-by-step integration method.

Three original contributions have been proposed in this book, namely, the principle of total potential energy with a stationary value in elastic system dynamics, the set-in-right-position rule for assembling system matrices, and the method of computer implementation in Matlab. Moreover, this book introduces the fundamental concepts of structural dynamics in a concise way rather than with a detailed description, which is more efficient for abecedarians in understanding the basic concepts and methods of vibration analysis.

Participants in the writing of this book include Zhihui Zhou, Ying Wen, Chenzhi Cai, and Qingyuan Zeng from Central South University. The specific division of the organization and writing of this book is as follows: Zhihui Zhou is responsible for the writing of Chapters 1 to 4; Ying Wen has fulfilled Chapters 5 and 6; Chenzhi Cai has completed Chapter 7, and Qingyuan Zeng supplied the original manuscript of the book.

The authors wish to express their sincere thanks and appreciation to Prof. Xiaojun Wei from Central South University, Prof. Tong Qiu from The Pennsylvania State University, and PhD student Juanya Yu from University of Illinois at Urbana-Champaign for valuable advice in the process of writing. The authors are also grateful to Mr. Lican Xie, Ms. Manxuan Yang, Mr. Liang Zhang, Mr. Bao Zhang, Mr. Xuanyu Liao, Mr. Chenlong Tang, Mr. Zhenhua Jian, Mr. Xiaojie Zhu, and other graduate students from Central South University for their contributions in different ways to the content of this book.

This book can be used as a textbook for both postgraduates and undergraduates majoring in civil engineering, engineering mechanics, mechanical engineering, and other related fields in general colleges and universities. It can also be a reference for teachers, general students, and short-term trainees in institutions of higher vocational education.

The authors cordially invite the audience of this book to contact with us (Zhihui Zhou: [e-mail]) if you have any suggestions for improvements and clarifications in the content organization, and even to help identify errors. All the above efforts and comments are sincerely acknowledged.

Chapter 1

Overview of structural dynamics

Abstract

This is an introductory chapter for structural dynamics in this book. Based on the dynamic examples in practical engineering, the objective of structural dynamic analysis is introduced. Three characteristics of structural dynamics, as distinct from static problems, are investigated, including time-varying property, effect of inertial force, and damping force. Four types of classification of vibrations in engineering are introduced: deterministic or random vibration, linear or nonlinear vibration. Vibration problems in engineering can be classified into response analysis, environment prediction, system identification, and system design. This book will focus on the dynamic response analysis which is the basis of structural dynamics. Procedures of dynamic response analysis of structures are discussed.

Keywords

Characteristics of dynamics; classification of vibrations; dynamic response analysis; deterministic vibration; random vibration

1.1 Objective of structural dynamic analysis

Dynamic analysis of the train–bridge system originated from the collapse of the Chester Railway Bridge in the United Kingdom due to a train passing over the bridge. In November 1940 the engineering community was astonished by the dynamic instability of the Tacoma suspension bridge in the United States under strong wind with a speed of 17–20 m/s. A large crowd of people participated in the opening ceremony of Wuhan Yangtze River Bridge in 1957, resulting in continuous swaying of the newly opened bridge. The swaying came to an end when the crowd went away at night. In 2011 the administrator of the Shanghai Railway observed the excessive transverse vibration of the Nanjing Yangtze River Bridge under the condition of a cargo train passing over the bridge. The transverse amplitude of the oscillated bridge exceeded 9 mm, which led to concerns over the safety of running trains on the bridge. Therefore the assessment of the safety and comfort of running trains on this bridge was conducted [1,2].

Seismic activity has been relatively active in recent decades, for instance, the Chilean earthquake in 1960, the Tangshan earthquake in China in 1976, the Mexico earthquake in 1985, the Osaka–Kobe earthquake in Japan in 1995, the India earthquake in 2001, and the Sichuan earthquake in China in 2008. In addition to serious disruption to the local economy, these disasters threatened the safety of residents and their properties in the concerned areas. Thus the aseismic design of infrastructures in seismically active areas is necessary to reduce or avoid severe earthquake damage for major projects. In addition, many airplane accidents have been caused by the flutter of aircraft wings or the abnormal vibration of engines. In mechanical engineering, vibrations may bring about negative effects on the performance of some precision instruments, for instance, these vibrations may increase abrasion and fatigue, or reduce machining accuracy and surface finish. However, some manufacturing facilities, for example, transmission, screening, grinding, piling, and so on, as well as various generators and clocks, benefit from the positive aspects of vibrations [3].

The investigation of structural dynamics focuses on understanding the basic mechanism of vibrations and presenting the corresponding processing methods. These methods can be adopted to eliminate of the negative vibration effects of machines, prevent dynamic instability of bridges and improve the tamping and compaction performances of the road construction machinery, and so on.

1.2 Characteristics of structural dynamics

The main differences between statics and dynamics can be addressed in the following aspects: (1) in dynamics, both the loads and responses of structures are time-varying, which implies that, unlike static problems, the solution of dynamics cannot be a single one. Therefore the dynamic analysis of structures presents a more complex and time-consuming process when compared with the static analysis of structures; (2) acceleration is significant in dynamics. The so-called inertial force produced by acceleration acts in the opposite direction of the acceleration. As illustrated in Fig. 1.1A, the internal moment and shear of the cantilever beam should equilibrate the applied dynamic load, F(t), as well as the inertial force associated with the acceleration. In Fig. 1.1B, the internal moment, shear, and deflection of the cantilever beam under a static load F depend only on the applied load itself. In general, once the inertial force accounts for a relatively large proportion of the forces equilibrated by the elastic internal force, the dynamic characteristics should be taken into account in the structural analysis. When applied loads do not change significantly, the dynamic responses are minor and the inertial forces can be neglected. Thus the static analysis procedure could be applied at any desired instant of time in these cases. If the exciting frequency is less than one third of the first natural frequency of the structure, the analysis of the structure could be treated as a static problem (a better understating of this concept can be achieved by means of Fig. 3.14); (3) damping is also an indispensable factor in dynamic problems. Energy will be dissipated in the vibration of structures. Structural damping is frequently ignored in the analysis of the natural dynamic properties and the dynamic response over a relatively short duration (such as the action of impulsive loads). However, structural damping must be taken into account when large damping exists or vibration lasts a long period, as well as in the analysis of the vibration in the resonance region.

Figure 1.1 Cantilever beam subjected to (A) dynamic load and (B) static load.

1.3 Classification of vibrations

1. The vibrations could be classified as either deterministic or random vibrations according to the deterministic or random characteristics of the dynamic responses.

a. Deterministic vibration: the structural responses are deterministic functions of time due to the determined load and system.

b. Random vibration: the structural responses are random due to the uncertainty of load or system. However, the responses usually comply with certain statistical rules and can be analyzed with statistical probability methods. For instance, the vibrations of aircraft owing to aerodynamic noise, the vibrations of the train–track–bridge system caused by track irregularity, etc., are all regarded as random vibrations.

2. The vibrations could be classified as either free vibrations, forced vibrations, self-excited vibrations, or parametric vibrations.

a. Free vibration: external perturbation makes the system deviate from the initial equilibrium position or have initial velocity. When the perturbation is rapidly removed, the system