Structural dynamics [Second edition] 9781119605775, 1119605776, 9781119605812, 1119605814

Citation preview

Structural Dynamics

Scrivener Publishing 100 Cummings Center, Suite 541J Beverly, MA 01915-6106 Publishers at Scrivener Martin Scrivener ([email protected]) Phillip Carmical ([email protected])

Structural Dynamics

Yong Bai Zhao-Dong Xu

This edition first published 2019 by John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA and Scrivener Publishing LLC, 100 Cummings Center, Suite 541J, Beverly, MA 01915, USA © 2019 Scrivener Publishing LLC For more information about Scrivener publications please visit www.scrivenerpublishing.com. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions. Wiley Global Headquarters 111 River Street, Hoboken, NJ 07030, USA For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Limit of Liability/Disclaimer of Warranty While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials, or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Library of Congress Cataloging-in-Publication Data ISBN 978-1-119-60560-7 Cover image: Seismic Wave | Sergey Khakimullin | Dreamstime.com High Rise Building | Scgonsalves | Dreamstime.com, Equations provided by Bai and Xu Cover design by Kris Hackerott Set in size of 11pt and Minion Pro by Manila Typesetting Company, Makati, Philippines Printed in the USA 10 9 8 7 6 5 4 3 2 1

Contents Preface About the Authors 1

Introduction 1.1 Overview of Structural Dynamics 1.2 Dynamic Loads 1.2.1 Simple Harmonic Loads 1.2.2 Nonharmonic Periodic Loads 1.2.3 Impulsive Load 1.2.4 Irregular Dynamic Load 1.3 Characteristics of a Dynamic Problem 1.3.1 Methods of Discretization 1.3.2 Lumped Mass Procedure 1.3.3 Generalized Coordinate Procedure 1.3.4 Finite Element Method 1.4 Application of Structural Dynamics 1.4.1 Application of Structural Dynamics in Civil Engineering 1.4.2 Application of Structural Dynamics in Ocean Engineering 1.4.3 Application of Structural Dynamics in Aircraft Technology Exercises References

2 Establishment of the Structural Equation of Motion 2.1 General 2.1.1 Dynamic Freedom 2.1.2 Basics of Dynamic System Inertia Force Elastic Restoring Force Damping Force

xi xiii 1 1 2 2 3 3 3 4 6 6 7 9 10 10 11 14 15 16 17 17 17 18 18 19 19 v

vi

Contents 2.2 Formulation of the Equations of Motion 2.2.1 Direct Equilibration Using D’Alembert’s Principle 2.2.2 Principle of Virtual Displacements 2.2.3 Hamilton’s Principle 2.2.4 Lagrange’s Equations 2.3 Theory of Total Potential Energy Invariant Value of Elastic System Dynamics 2.3.1 The Main Idea of the Principle of Virtual Work 2.3.2 Derivation of the Principle of Total Potential Energy Invariant 2.4 Influence of Gravitational Forces 2.5 Influence of Support Excitation Exercises References

21 21 23 26 30

3 Single Degree of Freedom Systems 3.1 Response of Free Vibrations 3.1.1 Undamped Free Vibrations 3.1.2 Damped Free Vibrations 3.1.3 Damping and Its Measurement 3.2 Response to Harmonic Loading 3.2.1 Harmonic Vibration of an Undamped System 3.2.2 Harmonic Vibration of Damping System 3.2.3 Dynamic Amplification Coefficient 3.2.4 Resonance Reaction 3.2.5 Solution of Damping Ratio 3.3 Periodic Load Response 3.4 Impulsive Loading Response 3.4.1 Sine-Wave Impulse 3.4.2 Rectangular Impulse 3.4.3 Triangular Impulse 3.5 Response of Arbitrary Load 3.5.1 Duhamel Integral (Time-Domain Analysis) 3.5.2 Fourier Transform (Frequency-Domain Analysis) 3.6 Energy in Vibration 3.6.1 Energy in Free Vibration 3.6.2 Energy Dissipation of Viscous Damped System 3.6.3 Equivalent Viscous Damping 3.6.4 Complex Damping 3.6.5 Friction Damping 3.7 Structural Vibration Test

41 41 43 46 52 57 57 62 65 68 70 74 80 80 82 84 89 89 95 97 97 99 100 103 106 106

32 32 34 37 38 39 40

Contents vii 3.7.1 Introduction to Vibration Test 3.7.2 Exciting Equipment 3.7.3 Vibration Measuring Instrument 3.7.4 Data Acquisition and Analysis System 3.8 Vibration Isolation Principle 3.8.1 Active Vibration Isolation 3.8.2 Passive Vibration Isolation 3.9 Structural Vibration Induced Fatigue 3.9.1 Definition of Vibration Induced Fatigue 3.9.2 Characteristics of Vibration Induced Fatigue Exercises References 4 Multi-Degree of Freedom Systems 4.1 Two Degrees of Freedom System 4.1.1 Establishment of Motion Equation of Undamped Free Vibrations 4.1.2 Natural Frequency and Vibration Mode Shape 4.1.3 General Solutions of the Equations of Motion 4.2 Free Vibrations of Undamped System 4.2.1 Establishment of Motion Equation 4.2.2 Vibration Shape and Its Orthogonality 4.2.3 Generalized Mass and Generalized Stiffness 4.3 Practical Calculation Method of Dynamic Characteristics 4.3.1 Dunkerley Formula 4.3.2 Rayleigh Energy Method 4.3.3 Ritz Method 4.3.4 Matrix Iteration Method 4.3.5 Subspace Iteration Method 4.4 Mode Superposition Method for Damped System 4.4.1 Coordinate Coupling and Regular Coordinates 4.4.2 Damping Assumptions 4.4.3 Mode Superposition Method 4.5 Numerical Analysis of Damping System 4.5.1 Central Difference Method 4.5.2 Average Constant Acceleration Method 4.5.3 Linear Acceleration Method 4.5.4 Newmark-β Method 4.5.5 Wilson-θ Method 4.6 Stability and Accuracy Analysis of Stepwise Integration Method

106 107 110 114 114 114 116 121 121 122 123 125 127 128 128 131 134 135 135 137 142 146 147 150 156 160 167 172 173 174 179 185 186 187 191 193 195 199

viii

Contents 4.6.1 Stability Analysis of Algorithm Solutions 4.6.2 Accuracy Analysis of Algorithm Solutions Exercises References

202 202 203 205

5 Distributed-Parameter System 5.1 Overview 5.2 Establish Differential Equations for Motion 5.2.1 Euler-Bernoulli Beam 5.2.2 Beam with Axial Pressures 5.2.3 Beam Flexure with Viscous Damping 5.2.4 Beam Axial Deformations without Damping 5.3 Free Vibration of a Beam 5.3.1 Decoupling the Boundary Conditions 5.3.2 Simply Supported Beam 5.3.3 Free-Free Beam 5.4 Orthogonality Relationships 5.5 Modal Decomposition References

207 207 208 208 210 211 211 213 214 215 217 221 223 225

6 Stochastic Structural Vibrations 6.1 Overview 6.2 Stochastic Process 6.2.1 Concept of Stochastic Process 6.2.2 Probability Description of Stochastic Processes 6.2.3 The Numerical Characteristics of Stochastic Processes 6.2.4 Stationary Stochastic Process 6.2.5 Several Important Stochastic Processes 6.2.6 Stochastic Model of Seismic Ground Motion 6.3 Stochastic Response of Linear SDOF System 6.3.1 Time-Domain Analysis Method 6.3.2 Frequency-Domain Analysis Method 6.3.3 Cross-Correlation Function and Cross-Spectral Density of Excitation and Response 6.3.4 Fatigue Predictions for Narrowband Systems 6.4 Stochastic Response of Linear MDOF System 6.4.1 Direct Method 6.4.2 Vibration Mode Superposition Method 6.5 Nonlinear Structural Stochastic Response Analysis 6.5.1 Perturbation Method 6.5.2 Equivalent Linearization Method

227 227 230 230 232 234 248 251 253 260 260 263 266 270 271 272 280 291 292 294

Contents ix 6.6 State Space Method for Structural Stochastic Response Analysis 6.6.1 Basic Concept of State Space 6.6.2 SDOF System 6.6.3 MDOF System Exercises References 7 Research Topics of Structural Dynamics 7.1 Analysis of Structural Seismic Response 7.1.1 Brief Introduction to the Calculation Method 7.1.2 Horizontal Seismic Action of SDOF Elastic System 7.1.3 Seismic Response Spectrum 7.1.4 Vibration Mode Decomposition Method 7.1.5 Bottom Shearing Force Method 7.2 Structural Vibration Control 7.2.1 Concept and Classification 7.2.2 Vibration Reduction Technology of Viscoelastic Dampers 7.2.3 Rubber Base Isolation Technology 7.2.4 Vibration Reduction Technology of Magneto-Rheological Damper 7.3 Modal Analysis and Theory 7.3.1 Modal Parameters 7.3.2 Real Modal Analysis 7.3.3 Complex Modal Analysis 7.4 Structural Dynamic Damage Identification 7.4.1 Frequency Base Damage Identification Method 7.4.2 Modal Base Damage Identification Method 7.4.3 Damage Identification Method Based on Stiffness Variation 7.4.4 Damage Identification Method Based on Flexibility Change 7.4.5 Energy-Based Damage Identification Method 7.4.6 Prospects for Research on Dynamic Damage Identification 7.5 Nonlinear Problems of Dynamic Analysis 7.5.1 Physical Nonlinearity Problems in Dynamic Analysis 7.5.2 Geometric Nonlinearity Problems in Dynamic Analysis 7.6 Sub-Structure Method

297 298 299 302 304 304 305 305 307 308 310 314 317 323 323 325 332 337 341 342 344 345 350 350 351 354 355 356 357 358 359 362 365

x

Contents 7.6.1 Finite Element Analysis of Sub-Structure Method 7.6.2 Damage Identification by Sub-Structure Method 7.7 Dynamics of Offshore Structures 7.7.1 Descriptions of Offshore Waves 7.7.2 Introduction to Wave Spectra 7.7.3 Frequency Domain Analysis Exercises References

365 368 369 370 370 371 373 373

8 Structural Dynamics of Marine Pipeline and Riser 8.1 Overview 8.2 Environmental Conditions 8.2.1 General 8.2.2 Linear Wave Theory 8.2.3 Nonlinear Wave Theory 8.2.4 Current 8.3 Hydrodynamic Loads 8.3.1 Hydrodynamic Drag and Inertia Forces 8.3.2 Hydrodynamic Lift Forces 8.4 Structural Response Analysis 8.4.1 Global Deformation Due to Environmental Loads 8.4.2 Mass Matrices 8.4.3 Stiffness Matrices 8.4.4 Damping Matrices 8.4.5 Riser Deformation 8.5 Vortex Induced Vibrations 8.5.1 Introduction 8.5.2 Analysis of Vortex-Induced Vibration 8.5.3 Harmonic Model 8.5.4 Wake Oscillator Model Exercises References

375 375 376 376 377 384 384 386 386 390 392 392 394 397 399 400 401 401 404 406 409 415 415

Answers to Exercises

417

Index

443

Preface Dynamic problems of structures are ubiquitous in research. Therefore, it is very important for students majoring in civil engineering, mechanical engineering, aircraft engineering and ocean engineering to systematically grasp the basic concepts, calculation principles and calculation methods of structural dynamics. This book focuses on the basic theories and concepts, as well as the application and background of theories and concepts in engineering. Since the basic principles and methods of dynamics are applied to other various engineering fields, this book can also be used as a reference for undergraduate and graduate students in other majors. The main contents include basic theory of dynamics, establishment of equation of motion, single degree of freedom systems, multi-degree of freedom systems, distributed-parameter systems, stochastic structural vibrations, research projects of structural dynamics, and structural dynamics of marine pipeline and riser. This book was co-authored by Professor Yong Bai of Southern University of Science and Technology and Professor Zhao-Dong Xu of Southeast University. The authors would like to appreciate Dr. Yong Bai’s and Dr.  Zhao-Dong Xu’s graduate students and postdoctoral fellows who provided the initial technical writing. The students in Southern University of Science and Technology are Ms. Xinyu Sun (Chapters 1 & 3), Mr. Jiannan Zhao (Chapters 2 & 4), Mr. Zhao Wang (Chapters 5, 6, & 7), and Mr. Wei Qin (Chapter 8). The students in Southeast University are Mr. Yanwei Xu (Chapter 1 and proofread all), Mr. Hao Hu (Chapter 3), Mr. Yun Yang (Chapter 6), Mr. Shi Chen (Chapter 7), and Mr. Qiangqiang Li (proofread all). The students in Xi’an University of Architecture and Technology are Mr. Zefeng He (Chapter 3), Mr. Zhenhua He (Chapter 4), and Ms. Tian Zhang (proofread all). Thanks to all persons involved in reviewing the book.

xi

About the Authors Professor Bai received a doctorate from Hiroshima University in Japan and engaged in postdoctoral work in the field of ocean engineering in Technical University of Denmark, Norwegian University of Science and Technology and University of California at Berkeley. He has published over 100 research papers, 9 English academic treatises and 8 Chinese books on Ocean Engineering. Bai served as a professor at University of Stavanger, Harbin Engineering University, Zhejiang University and Southern University of Science and Technology. He guided more than 50 graduate students and 30 doctoral students. Professor Bai has a wealth of engineering experience and management skills. He worked in Det Norske Veritas, American Bureau of Shipping, JP KENNY Company in Norway, Shell E & P Company and MCS in the United States. He has presided over dozens of large projects in the field of ship structures, submarine pipelines and risers, design analysis and risk assessment of offshore platforms. Bai put forward the design concept of buckling and ultimate load carrying capacity of deepwater submarine pipelines. He improved the design methods, analytical tools and design standards of marine pipelines and reached the international leading level. He significantly improved the design methodology and criteria for subsea pipelines and risers such as ultimate strength design, use of risk and reliability methods. He contributed to subsea technology by publishing many papers and a recognized book entitled Subsea Engineering Handbook and promoted limit-state design and use of risk and reliability by teaching at universities and publishing a book entitled Marine Structural Design. Professor Zhao-Dong Xu is the professor at the Civil Engineering School of Southeast University, serving as doctoral tutor. His major research fields are Anti-earthquake of Structures, Structural Control and Health Monitoring, Smart Material and Structures. Professor Xu got his Ph.D. in China, followed by a series of teaching and research positions at Xi’an Jiaotong University, Ibaraki University, North Carolina State University xiii

xiv

About the Authors

and University of Illinois at Urbana-Champaign. He is the Vice President of RC & PC Key Laboratory of Education Ministry. He has also been Changjiang Scholar Distinguished Professor and the National Science Fund for Distinguished Young Scholars in China. Professor Xu engaged in teaching and research on structural dynamics for more than 20 years. He has published more than 200 papers on the subject of structure dynamics research, numerical analysis and application of civil engineering, etc. He has been honored with many awards—the 43rd Geneva International Patents Exhibition Gold Award, the Second Award of National Award for Technological Invention in China, the Top Award of Chinese Building Materials Technology Invention, etc. He has completed many significant research projects in the areas of structural vibration control and structural health monitoring, and many research outcomes have been utilized in major real applications.

1 Introduction 1.1 Overview of Structural Dynamics Have you ever thought about the technology used in the Shenzhou spacecraft that we are all so proud of? Have you ever thought about what kind of marvelous power it takes to make planes which weigh tons fly while we enjoy them? What’s the reason for the collapse of Tacoma Narrows Bridge when it suffered 19m/s wind? Why are buildings with seismic resistance and isolation technology considered better in terms of seismic safety? All those subjects exist in nature, and are aspects of the subject of advanced dynamics as well. The theoretical study of dynamics began in the seventeenth century, and the publication of Analytical Mechanics by Joseph Louis Lagrange (1736– 1813) laid the foundation for the dynamic analysis in the linear system. With the development of science and technology, a variety of dynamic dives are applied to different engineering structures, which allows the theory of structural dynamics to move forward constantly. Up to now, we can already accomplish the dynamical analysis for huge complex structures with thousands of freedom degrees. With regard to the design or analysis for a structure, static problems are always major areas which should be of primary concern. However, a structure often comes to failure when critically subjected to dynamic loading. Structural dynamic analysis thus frequently plays as the control function in a structure’s design, which may be far more critical than static load for the damages of a structure. Examples include seismic-induced structure collapse, wind-induced failure of bridges or other long-span flexible structures, deformation of pile and destabilization of foundation under impact loads. Thus, it’s indispensable to conduct the dynamic analysis for engineering structures’ study, design, and security evaluation. Despite the fact that numerous pseudo-static calculation methods are adopted in

Yong Bai and Zhao-dong Xu (eds.) Structural Dynamics, (1–16) © 2019 Scrivener Publishing LLC

1

2

Structural Dynamics

some specifications for structural design and structural dynamic analysis for simplicity, such as the response spectrum method in seismic design code or the equivalent statics wind stress which is used to substitutes the actual wind stress in wind-resistance design, their theoretical basis is still from structure dynamics. Hence, it’s still essential to conduct the dynamic analysis in these solution procedures, such as solving the structure’s natural period and modes in multi-degree-of-freedom system, all of which are necessary parameters involved in response spectrum method. Structural dynamics is a theoretical and technical subject to study the dynamic characteristics of structural systems (mainly referring to the period, frequency, mode, and damping characteristics) and determine the dynamic responses of structures under dynamic loads (including internal force, strain, displacement, speed, acceleration etc.). The fundamental purpose of this discipline is to provide a solid theoretical basis for improving the safety and reliability of engineering structural systems in the dynamic environment.

1.2 Dynamic Loads According to whether a load is time-varying or not, loads are divided into dynamic load and static load. For static load, its magnitude, direction, and action point of static loads don’t change or change with time slightly, such as dead weight of structure, snow load, ash load, etc. On the other hand, dynamic loads change with time. In addition, dynamic loads will also bring structure inertial forces, which cannot be neglected and must be taken into consideration. Typical dynamic loads include simple harmonic oscillation caused by working machinery, wind loads, seismic loads, etc.

1.2.1 Simple Harmonic Loads Simple harmonic loads are the loads varying with time harmonically and periodically, which can be represented by harmonic function, such as P(t)=P0sinθt and P(t)=P0cosθt. Analyses of structural response under the simple harmonic loads are of great importance, not only because these dynamic loads actually exist in engineering structures (centrifugal load caused by cam axial rotation), but also any nonharmonic periodic loads can be represented as a sum of a series of simple harmonic components. Thus, in principle, structural dynamic response caused by any periodic loading can be translated into a superposition of responses created by a series of simple harmonic components. Furthermore, the responses of a

Introduction 3 Dynamic loads

Prescribed loads

Periodic loads

Random loads

Nonperiodic loads

Simple harmonic loads

Impulsive loads

Nonharmonic loads

Other regular loads

Wind loads

Seismic force loads

Other nondeterministic loads

Figure 1.1 Classification of dynamic loads.

structure under simple harmonic loads can reflect its dynamic characteristics; therefore simple harmonic loads play a vast role in the structural dynamic analysis.

1.2.2 Nonharmonic Periodic Loads Nonharmonic periodic loads are periodic functions of time, which vary with time periodically. They are different from simple harmonic functions. Examples include the hydrodynamic pressure of calm waves on dams and thrust generated by a propeller of a ship.

1.2.3 Impulsive Load Magnitude of impulsive load can increase or decrease rapidly in short duration, for example, impact load produced by explosion or blast.

1.2.4 Irregular Dynamic Load Irregular dynamic loads are difficult to be expressed by analytic expression because of the complexity and arbitrariness of its magnitude, direction, and position, such as earthquake action or wind load acting on the structure. Four classifications of loads are shown in Figure 1.2.

4

Structural Dynamics

P(t) t

Rotating mechanical loads

Simple harmonic load

P(t)

t

The driving force of the boat

Non harmonic periodic

P(t)

t

Explosion impulsive wave

Impulsive loads

P(t)

t

Earthquake suffered by the water tower

Random loads

Figure 1.2 Types of dynamic load.

1.3 Characteristics of a Dynamic Problem A structural dynamic problem is intended to solve for the response of the structure under dynamic loads, which differs from its static loading counterpart in the following two important aspects.

Introduction 5 The first is the time varying nature of the dynamic problem. Because the dynamic loads vary with time, the analyst must calculate a succession of solutions corresponding to all times in the response history when computing the dynamic responses of structure. Thus, a dynamic analysis is clearly more complex and time-consuming than a static analysis. Secondly, the inertia force must be considered in the dynamic problem. Compared with the static problems, the inertial force brought in by acceleration due to rapid variation of displacement in the structural dynamic reaction will seriously influence the structural dynamic response, and the direction of inertia force oppose the direction of acceleration. If a simple beam is subjected to a static load, F, as shown in Figure 1.3, forces acting on the simple beam are only external force, F, and support reactions. However, if F is a dynamic load, displacement of the beam will change rapidly. Therefore, in addition to external force F and support reactions, there is also an inertial force distributed along the beam’s central axis acting on the simple beam. Magnitude of the inertial force depends on motion of the beam, which is significantly influenced by the inertial forces themselves. Occurrence of inertial force makes analysis of structure responses more complex. Especially when the loading rate gets faster, the additional responses induced by inertial force may be far bigger than the corresponding responses caused by static force. Dynamic calculations must be conducted in all time domain, while inertia force induced impact must be taken into consideration. Occurrence of inertial force makes analysis more complex, but understanding and effective treatment can significantly simplify the complexity of dynamic analysis. Structural dynamics differs from statics due to the inclusion of inertial force, in other words, taking the influence of inertia force into consideration. Whether an engineering structure is analyzed as a dynamic problem depends on whether the load provokes a large acceleration response. If the acceleration response is slight, the inertial forces merely represent a smaller portion of the total load equilibrated by the elastic forces of the structure, F

F(t)

Inertial force (a) Static problem

Figure 1.3 The difference between static and dynamic.

(b) Dynamic problem

6

Structural Dynamics

there will be no significant distinction between effects of dynamic and static, and these kinds of problems can be solved by static structural analysis procedures.

1.3.1 Methods of Discretization The inertia force is the essential cause of creating structure dynamic response, which is resulted from structure mass. Thus, describing the position and motion of the structural mass for a structure becomes the key factor in structural dynamic analysis procedure, which results in a different definition for structural degree of freedom (DOF) in structural dynamics and statics. DOF in dynamics is defined as the number of requisite independence parameters for ascertaining the spatial mass locations of system in dynamical analysis. These independence parameters are also called generalized coordinate, which can be displacement, deflection, or other generalized quantities. Generalized coordinate and dynamic DOF will be presented in Chapter 2. Selecting a correct dynamic freedom is of great importance to describe inertial force of system accurately. All structural systems have a distributed mass, and all of them have an infinite number of degrees of freedom. However, it is extremely difficult to conduct calculations on an engineering structure with infinite DOFs, except some simple structures can be dealt with like a construction with infinite DOFs. In structural dynamics analysis, structures are usually simplified as a finite DOFs system to avoid the mathematical complexity in the procedure of calculation, which is called the procedure of discretization for a structure. The common discretization methods include lumped mass procedure, generalized displacements procedure, and finite element procedure. By all accounts, discretization method is the way of transforming an infinite DOFs problem into a finite DOFs one.

1.3.2 Lumped Mass Procedure Lumped mass procedure is the most commonly used in structural dynamic analysis, which centralize the quality of the actual structure in some geometric points according to some certain rules, while the structural bars are massless. In this way, an infinite DOFs system will be translated into a finite DOFs one. Figure 1.4 shows two examples of converting an infinite DOFs system into a finite degree of freedom system by the lumped mass procedure for the structures with continuous distribution mass. Figure 1.4a shows the

Introduction 7 m1

m

m2

m3

(a) Simple u3(t)

m3

m2

m1

u2(t)

u1(t)

(b) Frame

Figure 1.4 Schematics of lumped mass procedure.

procedure of translating a simple beam into a finite degree of freedom with three masses by lumping its continuous distribution mass m at three discrete points on the beam, that is, replacing the continuous distribution mass with lumped masses m1, m2, and m3. If taking the transverse movement in the plane of beam into consideration, the lumped mass beam has three transverse degrees of freedom. Figure 1.4b shows a plane frame structure with three stories, if we centralize half mass of columns, walls, floor, beam of every contiguous stories at the center of their floor. The three stories frame structure will be simplified as a finite DOFs system with three lumped mass points.

1.3.3 Generalized Coordinate Procedure Generalized coordinates are the independent parameters which can determine the geometric position of the system. It can be adopted in the beam system with the distribution mass to improve calculation accuracy. Series method is normally used to solve differential equations in the field of mathematics, which can also be used to solve problems in structure dynamic

8

Structural Dynamics

analyses. As shown in Figure 1.5, the deformation curve of a sample beam with distribution mass can be expressed as a sum of the Fourier series. N

y( x , t ) i 1

n x bi sin L

N

bi (t )sin i 1

n x L

(1.1)

Where L is the length of beam; sin iπx/L is shape functions, it is a series of given functions that satisfy the boundary condition; bi = bi(t) represent generalized coordinates, it is a set of parameters to be determined, and it is a function of time in dynamics. Because the shape function is a predefined and ascertained function, the deformation of the beam will be determined by multiple generalized coordinates, this makes the simple beam have infinite degrees of freedom in the theoretical dynamic analysis. The same approaches are taken as in mathematics, only the first few items of the series will be brought into the actual analysis. Its first N items are taken in the following example. N

y( x , t )

bi (t )sin i 1

n x L

(1.2)

Thus, a simple beam is simplified as an N degrees of freedom system. A more generalized expression of structure displacement may be written as: N

y( x , t )

qi (t ) i ( x ) i 1

qi (t ) i ( x )

(1.3)

i 1

y m

x

y(x)

Figure 1.5 Discretization of a simple beam by generalized coordinate procedure.

Introduction 9 Where qi(t) is amplitude of shape function, that is, generalized coordinate; φi(x) is shape function and it is a continuous function that satisfies the boundary condition, take the simple beam shown in Figure 1.5 for instance, the boundary condition that shape function must satisfy is φn(0) = φn(L) = 0. If the shape function represents displacement, the generalized coordinate will have the dimension of displacement. The generalized coordinates can represent the amplitude of the shape function, but it is not the actual physical displacement, because it cannot convey the real displacement unless all the N items are added.

1.3.4 Finite Element Method As with the finite element method (FEM) used in the static problem, structural dynamics problems can also be solved by the FEM which combines certain features of both the lumped mass and the generalized coordinate procedures. The procedure of FEM applied in dynamics analysis uses the displacements of structure nodal points to represent the displacement state of each point in the structure. Firstly, the structure would be divided into a series of units, and these units are connected with nodal points, then the displacements of nodal points are independent coordinates which determine the displacements of all mass point in structure. In discretization procedure of FEM, the assumption deflections of elements are added to obtain the deflection curve of the entire beam at a certain time. These deflection curves used to describe the shape of each units are described as displacement function or interpolation function, whose expression contains several parameters. These displacement functions should be smoothly continuous inside each element and satisfy the support and deformation continuity conditions at two ends of each element. According to these conditions, the parameters in the displacement function can be expressed by the nodal displacement. Thus, the entire structure system is transformed into a finite DOF system with unknow nodal displacement. Take a continuous beam, for instance; it can be divided into N elements (beam segments), the intersections of interconnected elements are called nodal points, and the parameters of nodal displacement (displacement u and rotation angle ) are taken as generalized coordinates. This finite element model has six generalized coordinates (displacement parameters): u1, θ1, u2, θ2, u3,θ3, the corresponding shape function is φ1, φ2, φ3, φ4, φ5, φ6.

10

Structural Dynamics

For a cantilever beam discretization model with N elements, there are 2N generalized coordinates in total, and the displacement of the beam can be described by 2N generalized coordinates and their shape functions, whose expression is shown below.

u( x )= u1 1 ( x )+

1

2

( x )+

+ uN

2 N -1

( x )+

N

2N

(x )

(1.4)

In this way, an infinite DOF beam can be transformed into a system with 2N finite DOF. The finite element procedure combines the characteristics of both the lumped mass and the generalized coordinate procedure: 1. Similar to the generalized coordinate procedure, the finite element procedure also uses the concept of shape function. Different from the interpolation (definition of shape function) in entire system (structure) in the generalized coordinate procedure, the finite element procedure adopts the piecewise interpolation method (definition of piecewise shape function), so the expression (shape) of the shape function can be relatively simpler. 2. Compared with the lumped mass procedure, the generalized coordinates of the FEM have real physical means too, the same as the lumped mass procedure, which has the advantages of being direct and intuitive. Among the above three procedures, the lumped mass procedure is relatively simpler and more practical, the generalized coordinate procedure needs to select suitable shape function to satisfy the displacement boundary condition, which is suitable for simple structures merely. The FEM combines the characteristics of the lumped mass procedure and generalized coordinate procedure, which is suitable for various complex structures and widely used to solve the dynamic problems of engineering structure indeed.

1.4 Application of Structural Dynamics 1.4.1 Application of Structural Dynamics in Civil Engineering When a building structure is subjected to earthquake, wind, vibration and other loads, the magnitude, direction and action point of such loads

Introduction 11 not only change with time, but also change greatly, which makes the load response of the structure very different from the static load. Therefore, this kind of load is defined as dynamic load. The essential difference between structural statics and dynamics is that the static load acts on the building structure to produce static internal force, deformation and other responses; the dynamic load acts on the building structure, the vibration makes it produce accelerated motion, the accelerated motion gradually attenuates due to damping effect. The dynamic load not only makes the structure deform; the inertia force caused by acceleration also makes the structure respond to internal force, deformation, motion, displacement, acceleration, and so on. Building structure is a space system that can be connected by several components. Building structures can be divided into concrete structures, masonry structures, steel structures, wood structures and composite structures according to the building materials used. Combined with functional characteristics, appearance design, planning and layout requirements, building structures are usually irregular in plane shape. When a plane irregular building structure is under stress, some irregular parts interact with each other, which makes the whole structure torsion. At the junction of irregular parts, stress concentration and deformation concentration are easy to occur. Since the wind load is less than the earthquake action, it is necessary to analyze the regularity lateral resistance of building structure plane according to the earthquake action, earthquake action + wind load action, dead load + live load + accidental load, indoor physical simulation test (jack test, shaking table test).

1.4.2 Application of Structural Dynamics in Ocean Engineering With the increasing demand for offshore oil and mineral resources, ocean engineering has developed rapidly in recent years. A large number of topics related to ocean engineering science need to be studied and solved. Marine structural dynamics mainly studies the interaction between waves and marine structures, the interaction between structures and seabed foundations. A reasonable and practical mathematical model, theory and method for structural analysis are proposed to solve the problems of structural vibration response, strength calculation and failure mechanism. A reasonable structural form is proposed for engineering design and construction. It provides the necessary theoretical basis and design parameters. Marine structural dynamics is an interdisciplinary subject which integrates structural dynamics, soil dynamics, geology, oceanography,

12

Structural Dynamics

mathematical statistics, computational methods, experimental techniques and other disciplines. It is a research direction with a wide range of disciplines and development prospects. The marine environment is very harsh, so human understanding of the ocean is very limited. Most of the offshore structures are submerged in sea water. The generation of structural load and the motion of the structure are closely related to the surrounding fluid medium. The dynamic loads of ocean structures can be divided into two categories: natural loads and artificial loads. According to the causes, natural loads can be divided into fluid loads, wind loads, earthquake loads and floating ice loads. Man-made loads refer to the dynamic loads generated in various production activities (including ship and machine impacts, drilling operations, machinery, crane operations, etc.). These loads are mainly fluid loads, which are to be discussed in this section. Earthquake and floating ice loads should be considered in some areas, which will be described separately below. Fluid loads are caused by waves, currents, eddy detachment, overshoot vibration, breaking and beating of waves, among which wave action is the main one. Waves are caused by wind blowing on the water surface and will continue to exist after wind stops. The shape of the wave surface is close to the ideal sinusoidal curve in deep water, and its amplitude is small. It tends to be in the same cycle in shallow water. The asymmetry of the wave surface increases with the increase of the steepness. There are two main factors indicating the wave characteristics, the period and the wave height. Degree propagates unidirectionally forward, but water points in steady waves move regularly at lower speeds. In a wave cycle, a water point completes a near-closed orbit with a vertical diameter equal to that height. In deep water, the trajectory of a water surface particle is close to a circle with a diameter varying with depth. When the depth exceeds one wavelength, the water point becomes almost static; in shallow water, the orbit of the surface particle is approximately elliptical, its long axis is parallel to the seabed, and its short axis decreases with the depth until the seabed has only horizontal motion. Waveform is caused by half of kinetic energy and comes from water point velocity. The energy density distribution with frequency is called spectrum, which will be used for spectral analysis of structures. Wave parameters are determined by wind speed, wind time and wind zone. Actual ocean waves are superimposed by waves of different periods and wave heights, which are caused by the changing size, range and direction. Therefore, ocean wind waves generally have broad spectrum characteristics and are suitable for spectral analysis. When the waves move to

Introduction 13 the generating area or when the wind stops. Surge propagates very far and keeps a certain shape over a long distance, and the wave with longer period has more components and narrow spectrum characteristics. Tides and winds produce currents at estuaries. Currents play a significant role below the surface of waves (about a wavelength area). Directional currents cause static loads and pulsating currents (turbulence) cause dynamic loads. Turbulence frequencies are low and have little effect on the dynamic analysis of buildings. Both of them can be derived from the sea according to the formulas of resistance and inertia force. The vortices are separated alternately from one side of the component in a steady flow, resulting in a vibration force perpendicular to the flow directionlift and upward pulsating resistance. The lift frequency is equal to the frequency of a pair of vortices detachment. The main frequency of the pulsating resistance is twice that of the lift frequency. The vortex detachment frequency, dimensionless lift coefficient and fluctuating drag coefficient depend on the section shape, relative flow direction and Reynolds number of the component. Over-ride vibration is another form of vibration that occurs in a structure in an ocean current. It is a self-excited vibration in which the force induced by the motion of the body is in its direction of motion. Thus, when the work of the fluid is done on the body, unstable vibration occurs when the input of the work is greater than the energy dissipated by the damping of the structure. In the case of directional jets, if the axis of the structural member is parallel to the sea surface, the component will alternately sink or expose to the water surface when the wave passes. The alternating impact loads can also cause the structure to be excited. Under the conditions of deep water and storm, the wave often reaches the ultimate steepness and breaks up, thus affecting the analysis of wave forces. Air entrainment will further complicate the problem. Direct breakage of waves on buildings may greatly exceed the wave forces normally calculated. These problems are still limited to qualitative simulation tests, and their mechanisms need to be further clarified. Earthquake and floating ice are special loads encountered by marine structures, which can be ignored under normal circumstances, but structures themselves in seismic or floating ice zones will become an important factor in the design load of marine structures. Load also has stochastic characteristics; it is due to the ground motion acceleration caused by the whole structure mass inertia force caused by dynamic response. Fatigue is an important problem in the analysis of marine structures. The reason is that the cyclic loads on marine structures are much more frequent than those on land. Under long-term cyclic loads, the cracks and

14

Structural Dynamics

their propagating members can also cause serious damage under low stress conditions. This is in accordance with the above maximum control. The first step is to calculate the stress response frequency, which is the combination of the natural frequency and wave frequency of the structure. It is often much higher than the wave frequency. In addition, it should be considered that the general local impact force can cause the movement of the component with its natural frequency. New attention is paid to the combinations of different loads. Some loads cause little stress amplitude, but the frequency is higher. There are also construction and installation dynamics problems for offshore structures. These problems are different from those under normal production conditions and require special consideration, including the dynamic response of floating stability, towing, throwing and sinking components in the course of transportation, the driving analysis of pipe piles, the local stress of caisson installation on non-uniform foundation, etc. In addition, there are a series of problems, such as foundation stability, soil liquefaction, material dynamic performance, dynamic stress concentration, dynamic plasticity, fracture, temperature difference, tsunami, landslide, uncertainty analysis and vibration tolerance standards.

1.4.3 Application of Structural Dynamics in Aircraft Technology Aircraft structures are often subjected to a variety of dynamic loads in use, resulting in structural dynamic strength failure. With the improvement of aircraft performance, it is more and more important to analyze the types and intensity of these vibration excitations and impacts. The degree of damage is not only related to the motion state and environment of the aircraft, but also closely related to the dynamic characteristics of the aircraft structure itself. Therefore, the dynamic strength of aircraft structure has become a very important issue in the process of aircraft design and decommissioning. In recent years, with the development of aircraft structural strength analysis and test technology, structural dynamics has made great breakthroughs in the application field of aircraft design, and its technical level has been continuously improved and improved. Some remarkable achievements and progress have been made in many aspects, such as aircraft complex structure dynamics analysis, dynamic test simulation and analysis technology, large structure vibration test design technology, formulation and application of mechanical environment conditions of airborne

Introduction 15 equipment, composite structure dynamics research, aircraft noise prediction and acoustic fatigue life analysis. However, up to now, structural dynamics is still a developing subject, so there are inevitably many problems to be further studied and discussed in the design of structural dynamics. For example, due to the complexity of vibration sources, the dynamic loads cannot be accurately described; because of the difficulty of simulating boundary conditions, the dynamic response analysis of large complex structures is not accurate enough; the damping characteristics of complex structures are still difficult to determine. Structural vibration induced fatigue failure mechanism, vibration induced fatigue life prediction method, dynamic strength criterion and criterion, etc. have become the key technical problems of future aircraft design and development. In the field of structural dynamic strength technology, some innovative engineering applications are brewing, mainly including: (1) Dynamics design technology of structures under dynamic loads, which is penetrating into modern design technology. (2) Structural adaptive vibration control or active control technology, which is developing and crossing between structural dynamics technology and modern control technology. (3) Structural vibration induced fatigue and dynamic fracture technology formed by the combination of structural dynamics technology and traditional fatigue technology. The development of these new technologies will greatly enrich and expand the theory of structural dynamics, and the application of these theories and new technologies to guide and solve the key problems of structural dynamics in aircraft design is an important task and goal of future aircraft structural dynamic strength work.

Exercises 1.1 What is the purpose of structural dynamics analysis? What are the characteristics of structural dynamics? 1.2 Express the types and characteristics of dynamic loads. 1.3 What are the procedures of structural discretization? What are their characteristics?

16

Structural Dynamics

References 1. Zhao-Dong, X. and Le-Wei, M., Dynamics of Structures, Science Press, Beijing, 2007. 2. Clough, R.W. and Penzien, J., Dynamics of structures, in: Dynamics of Structures, 2005. 3. Chopra, AK., Dynamics of Structures, Prentice Hall, 2000. 4. Wilson, J.F., Dynamics of Offshore Structures, 1984. 5. Forment, D. and Welaratna, S., Structural Dynamics Modification—An Extension to Modal Analysis. Sae Paper, 1980. 6. Herrada, F.J., García-Martínez, J., Fraile, A., Hermanns, L.K.H., Montáns., F.J., A method for performing efficient parametric dynamic analyses in large finite element models undergoing structural modifications. Eng. Struct., 131, 2017. 7. Kawabata, N. and Fukunaga, H., Identification of structural dynamic response based on the design of modal sensor. Proc. 1992 Ann. Meet. JSME/ MMD, 2003, 0, 2003. 8. Harte, M., Basu, B., Nielsen, S.R.K., Dynamic analysis of wind turbines including soil-structure interaction. Eng. Struct., 45, 2012. 9. Pasha, H.G., Kohli, K., Allemang, R.J. et al., Structural dynamics model calibration and validation of a rectangular steel plate structure, in: Model Validation and Uncertainty Quantification, vol. 3, Springer International Publishing, 2015. 10. Ozcelik, O., Attar, P.J., Altan, M.C. et al., Experimental and numerical characterization of the structural dynamics of flapping beams. J. Sound Vib., 332, 21, 5393–5416, 2013. 11. Herbert, M.R. and Kientzy, D.W., Applications of Structural Dynamics Modification, 1980.

2 Establishment of the Structural Equation of Motion The purpose of structural dynamics analysis is to calculate the dynamic response of the structure under dynamic load, that is, to solve the history of displacement, velocity, acceleration, strain, etc., of the structure over time. In most cases, applying an approximate analysis method with a finite number of degrees of freedom is accurate enough. In this way, the problem becomes to find the time-history curve of the selected component. Before solving the time-history curve, the equation of motion of the dynamical system under dynamic load must be established. This chapter will briefly introduce some of the basic concepts of structural dynamics and the methods for establishing the structural equations of motion.

2.1 General The degree of freedom is often talked about in structural dynamics, and it is necessary to be familiar with the concept of particles before describing degrees of freedom. Particles are ideal models for simplifying objects. The model is considered as objects which have only mass and no size.

2.1.1 Dynamic Freedom The number of independent geometric parameters required to describe the position of the system at any moment of mass during motion is called the number of degrees of dynamic freedom of the structure. The number of structural degrees of freedom is not fixed, changes as the structural calculation hypothesis changes for a structure. As shown in Figure 2.1, the single mass is depicted in Figure 2.1(a), and the mass which has two degrees of freedom can move in x-axis and y-axis; cantilever beam ignoring axial effect is shown in Figure 2.1(b), where the right mass can move in x-axis, and the left one can move in y-axis; a rigid beam is shown in Figure 2.1(c), if the stiffness is assumed to be infinite, whose three masses has single Yong Bai and Zhao-dong Xu (eds.) Structural Dynamics, (17–40) © 2019 Scrivener Publishing LLC

17

18

Structural Dynamics x y y x (a) Single Mass W=2

(b) cantilever beam W=2

θ

(c) rigid beam W=1

(d) four-layer frame W=4

Figure 2.1 Definition of degree of freedom.

degree of freedom, namely angle of rotation θ; the sketch of four-layered frame is shown in Figure 2.1(d), each mass can move in horizontal direction and the structure has four degrees of freedom. When analyzing a dynamic system, the first step is to determine the degree of power freedom and establish the differential equation of motion. Before describing how to establish the differential equation of motion of a dynamic system, it is necessary to understand the basic components of the dynamic system.

2.1.2 Basics of Dynamic System A dynamic system is a simple representation of physical systems and is modeled by mass, damping and stiffness. For any system subjected to a dynamic load, the main physical characteristics are the mass of the system, elastic recovery characteristics, energy dissipation characteristics or damping, and the external disturbance or load of the system.

Inertia Force Mass is a fundamental property of matter and is present in all physical systems. This is simply the weight of the structure divided by the gravity

Structural Equation of Motion Establishment 19 acceleration. Mass contributes an inertia force (equal to mass times acceleration) in the dynamic equation of motion, which can be expressed as,

FI (t )

mu t

(2.1)

where, FI(t) represents the inertia force; m represents the mass; ü(t) is the acceleration.

Elastic Restoring Force Stiffness makes the structure more rigid, lessens the dynamic effects and makes it more dependent on static forces and displacements. Usually, structural systems are made stiffer by increasing the cross-sectional dimension, making the structures shorter or using stiffer materials. Stiffness is the resistance it provides to deformations, mass is the matter it contains and damping represents its ability to decrease its own motion with time. Assuming that the relationship between the force and displacement is linear, the restoring force of the spring is also referred to as the elastic restoring force, which is equal to the product of the spring stiffness and displacement,

FS (t )

ku

(2.2)

where, FS (t) represents the restoring force; k represents the stiffness of the spring; u(t) is the displacement.

Damping Force Damping, in physics, is restraining of vibratory motion, such as mechanical oscillations, noise, and alternating electric currents, by dissipation of energy. Unless a child keeps pumping a swing, its motion dies down because of damping. Shock absorbers in automobiles and carpet pads are examples of damping devices. Whereas the mass and the stiffness are wellknown properties and measured easily, damping is usually determined from experimental results or values assumed from experience. There are several sources of damping in a dynamic system. Viscous damping is the most used damping system and provides a force directly proportional to the structural velocity. This is a fair representation of structural damping in many cases and for the purpose of analysis, and it is convenient to assume

20

Structural Dynamics

viscous damping (also known as linear viscous damping). For the single degree of freedom system, the viscous damping can be written as,

FD (t )

cu

(2.3)

where, FD(t) represents the damping force; c represents the damping coefficient; u(t ) is the velocity of mass. Viscous damping is caused by such energy losses as occur in liquid lubrication between moving parts or in a fluid forced through a small opening by a piston, as in automobile shock absorbers. The viscous-damping force is directly proportional to the relative velocity between the two ends of the damping device. Viscous damping is usually an intrinsic property of the material and originates from internal resistance to motion between different layers within the material itself. The motion of a vibrating body is also checked by its friction with the gas or liquid through which it moves. The damping force of the fluid in this case is directly proportional to a quantity slightly less than the square of the body’s velocity and, hence, is referred to as velocity-squared damping. Besides these external kinds of damping, there is energy loss within the moving structure itself that is called hysteresis damping or, sometimes, structural damping. In hysteresis damping, some of the energy involved in the repetitive internal deformation and restoration to original shape is dissipated in the form of random vibrations of the crystal lattice in solids and random kinetic energy of the molecules in a fluid. There are other types of damping. Resonant electric circuits, in which an alternating current is surging back and forth, as in a radio or television receiver, are damped by electric resistance. The signal to which the receiver is tuned supplies energy synchronously to maintain resonance. In radiation damping, vibrating energy of moving charges, such as electrons, is converted to electromagnetic energy and is emitted in the form of radio waves or infrared or visible light. In magnetic damping, energy of motion is converted to heat by way of electric eddy currents induced in either a coil or an aluminum plate (attached to the oscillating object) that passes between the poles of a magnet. Damping is the dissipation of energy from a vibrating structure. In this context, the term dissipate is used to mean the transformation of energy into the other form of energy and, therefore, a removal of energy from the vibrating system. The type of energy into which the mechanical energy is transformed is dependent on the system and the physical mechanism that cause the dissipation. For most vibrating system, a significant part of the energy is converted into heat.

Structural Equation of Motion Establishment 21

2.2 Formulation of the Equations of Motion As mentioned earlier, the primary objective of a deterministic structural dynamic analysis is the evaluation of the displacement time histories of a given structure subjected to a given time varying loading. In most cases, an approximate analysis involving only a limited number of degrees of freedom will provide sufficient accuracy; thus, the problem can be reduced to the determination of the time histories of these selected displacement components. The mathematical expressions defining the dynamic displacements are called the equations of motion of the structure, and the solution of these equations of motion provides the required displacement time histories. The formulation of the equations of motion of a dynamic system is possibly the most important, and sometimes the most difficult, phase of the entire analysis procedure. In this text, three different methods will be employed for the formulation of these equations, each having advantages in the study of special classes of problems. The fundamental concepts associated with each of these methods are described in the following paragraphs.

2.2.1 Direct Equilibration Using D’Alembert’s Principle D’Alembert’s principle, an alternative form of Newton’s second law of motion, was stated by the seventeenth-century French mathematician Jean le Rond d’Alembert. In effect, the principle transforms a problem in dynamics to a problem in statics. According to the second law, it states that the force F acting on a body is equal to the product of the mass m and acceleration ü of the body,

F mu

(2.4)

Equation (2.4), indicating that force is equal to the product of mass and acceleration, may also be written in the form

F mu 0

(2.5)

In other words, the body is in equilibrium under the action of the real force F and the fictitious force –mü. The fictitious force is also called inertial force and reversed effective force. Example 2.1 The dynamic system is shown in Figure 2.2, where the mass m is applied with the force P(t). Try to derive an equation of motion for the system.

22

Structural Dynamics u(t)

O k

P(t)

m c FI FS

m

FD

P(t)

Figure 2.2 Single body with SDOF dynamic system.

Solution: The body only moves along a horizontal direction. Therefore, the system is the single degree of freedom system. Assume the displacement of the body is u(t), the spring restoring force is Fs(t)=ku(t), and the damping force is FD cu(t ). The equilibrium equation under the external force P(t) can be written as,

mu t

ku t

cu t

P t

0

Further,

mu t

ku t

cu t

P t

Example 2.2 The dynamic system is shown in Figure 2.3, which is the dual body dynamic system. Try to derive an equation of motion for the system. P2

P1 k1

kg

k2 m1

m2

FI1 k1u1

FI2 k2(u2–u1)

m1

k2(u2-u1) P1(t)

Figure 2.3 Dual body with 2DOF dynamic system.

m2

k3u2 P2(t)

Structural Equation of Motion Establishment 23 Solution: The above system includes two particles with horizontal movement. These two particles can be analyzed separately as shown in Figure 2.3. For the mass m1, the real forces include the left spring force –k1u1, the middle spring force k2(u2-u1), and an external force P1(t). For the mass m2, the real forces include the middle spring force k2(u2-u1), the right spring force –k3u2, and an external force P2(t). Further, the equilibrium equations for two particles can be written as,

P1 t

k2 u2 t

u2 t

k1u1 t

m1u1 t

0

P2 t

k2 u2 t

u2 t

k2u2 t

m2u2 t

0

The matrix form can also be given as,

M u where, M K P u

K u

P

is the mass matrix is the stiffness matrix is the vector of the external force is the vector of the body displacement

D’Alembert’s principle is a very convenient device in problems of structural dynamics because it permits the equations of motion to be expressed as dynamic equilibrium. The force F may be considered to include many types of force acting on the mass: elastic constraints which oppose displacements, viscous forces which resist velocities, and independently defined external loads. Thus, if an inertial force which resists acceleration is introduced, the equation of motion is merely an expression of equilibration of all forces acting on the mass. In many simple problems, the most direct and convenient way of formulating the equations of motion is by means of such direct equilibrations.

2.2.2 Principle of Virtual Displacements However, if the structural system is reasonably complex involving a number of interconnected mass points or bodies of finite size, the direct

24

Structural Dynamics

equilibration of all the forces acting in the system may be difficult. Frequently, the various forces involved may readily be expressed in terms of the displacement degrees of freedom, but their equilibrium relationships may be obscure. In this case, the principle of virtual displacements can be used to formulate the equations of motion as a substitute for the direct equilibrium relationships. The principle states that if a system of mass particles which is in equilibrium under the action of a set of externally applied forces is subjected to any virtual displacement, i.e., a displacement pattern compatible with the system’s constraints, the total work done by the set of forces will be zero. N

Fi miui

ui

0

(2.6)

i 1

where i

is an integer used to indicate (via subscript) a variable corresponding to a particular particle in the system Fi is the total applied force (excluding constraint forces) on the i-th particle mi is the mass of the i-th particle üi is the acceleration of the i-th particle miüi together as product represents the time derivative of the momentum of the i-th particle δui is the virtual displacement of the i-th particle, consistent with the constraints

With this principle, it is clear that the vanishing of the work done during a virtual displacement is equivalent to a statement of equilibrium. Thus, the response equations of a dynamic system can be established by first identifying all the forces acting on the masses of the system, including inertial forces. Then, the equations of motion are obtained by separately introducing a virtual displacement pattern corresponding to each degree of freedom and equating the work done to zero. Example 2.3 The inverted pendulum system is shown in Figure 2.4. Try to establish the equation of motion for this system using virtual displacement principle.

Structural Equation of Motion Establishment 25 m Fm

a sin (φ1)

k

Fp Fk a cos (φ1)

l a

φ1 sin (φ1) ~ φ1 cos (φ1) ~ 1

O

l sin (φ1)

Figure 2.4 Inverted Pendulum system with SDOF.

Solution: In the tangential direction of the mass, we have the following forces, Fk = a•cos(φ1)k ≈a φ1 k

Spring force Inertia force

FI

1

•1• m

Fpsin(φ1) ≈ m•g•φ1

External force

The virtual displacement can be defined as shown in Figure 2.5,

δuk=δuk•a, δum=δul•l m

Fm Fp sin (φ1) Fk cos (φ1)

k

δuk

l a

δum

φ1 δφ1 sin (φ1) ~ φ1 cos (φ1) ~ 1

O

Figure 2.5 Inverted Pendulum system with 1DOF.

26

Structural Dynamics

According to the virtual displacement principle, the principle of virtual work can be derived as,

Fk cos

uk

1

Fm

Fp sin

um 0

1

Substituting the virtual displacement equation into Equation 2.6 and cancelling out δφ1 the following equation of motion is obtained,

m l2

a2 k m g l

1

1

0

The advantage of the principle of virtual displacement is that the virtual work contributions are scalar quantities and can be added algebraically, whereas the forces acting on the structure are vectorial and can only be superposed vectorially.

2.2.3 Hamilton’s Principle Hamilton’s principle is one of the most fundamental and important principles of mechanics and mathematical physics. For the non-conservative system of n particles, based on d’Alembert’s principle which states that, N

Fi mi i 1

d2ui dt 2

ui

0

(2.7)

Note that in this principle the knowledge of force, whether it is conservative or non-conservative, and also the requirement of holonomic or nonholonomic constraints, does not arise. We write the principle in the form, N

N

Fi i 1

ui

mi i 1

d2ui dt 2

ui

(2.8)

Now consider N

i 1

d2u mi 2i dt

N

ui = i 1

du d mi i dt dt

N

ui

mi i 1

dui d dt dt

Since we have du /dt = d( u)/dt, therefore, we write

ui

(2.9)

Structural Equation of Motion Establishment 27 N

i 1

N

d2u mi 2i dt

ui = i 1

du d mi i dt dt

ui

T

(2.10)

where T is the kinetic energy of the system

1 T= 2

N

i 1

2

dui mi dt

(2.11)

Substituting this in Equation 2.10, we get N

W= i 1

d du mi i dt dt

ui

T

(2.12)

Further N

W +T = i 1

d du mi i dt dt

ui

(2.13)

Integrating the above equation with respect to time interval between t0 to t1 N

t1

W +T dt= t0

i 1

du mi i dt

t1

ui

(2.14) t0

Since, there is no variation in co-ordinates along any paths at the end t1 points. i.e., ui t 0. Hence from above equation we have 0

t1

W +T dt=0

(2.15)

t0

This is known as Hamilton’s principle for non-conservative systems. If the system is conservative, then the forces are derivable from potential.

28

Structural Dynamics

In this case, it is helpful to separate the force vector F(t) into its conservative and non-conservative components as represented by

Ft

Fc t

Fnc t

(2.16)

A potential energy function, V(t), is then defined such that the conservative force vector Fc(t), by definition, must satisfy the component relations

Fci

ui

i

i

V ui ui

V

(2.17)

Hence equation 2.15 becomes t1

t1

T V dt+ t0

t0

Wnc dt=0

(2.18)

Upon considering a summation of equations of this type for all mass particles, it becomes apparent that equation 2.18 is also valid for any complicated system, linear or nonlinear, provided quantities T(t), V(t), and Wnc(t) represent the summation of such quantities for the entire system. Equation (2.18), which is generally known as Hamilton’s variational statement of dynamics, shows that the sum of the time variations of the difference in kinetic and potential energies and the work done by the nonconservative forces over any time interval t1 to t2 equals zero. The application of this principle leads directly to the equations of motion for any given system. Example 2.4 Use Hamilton’s principle to find the equations of motion of a particle of unit moving as shown below. O

u(t)

k m c

Figure 2.6 Single body with 1DOF dynamic system.

P(t)

Structural Equation of Motion Establishment 29 Solution: The kinetic and potential energy of the particle is given by

1 2 mu , V 2

T

1 2 ku 2

The work done by the nonconservative force can be given by

Wnc

P u cu u

The Hamilton’s principle states that t1

1 2 1 2 mu ku + P u cu u dt=0 2 2

t0

Further, t1

mu u ku u+P u cu u dt=0 t0

where t1

t1

mu udt= t0

mu t0

d u dt

t1

dt=mu u t

0

t1

mu udt t0

Since (mu u)tt10 0 , and δu is independent and arbitrary, then we have t1

t1

mu u ku u+P u cu u dt= t0

mu ku+P cu

udt=0

t0

Finally,

mu+cu+ku=P In summary, the specific steps for establishing the equation of motion using Hamilton’s principle are: 1) Explicitly study objects, analyze constraints, determine system’s degree of freedom and choose appropriate coordinates

30

Structural Dynamics 2) Calculate the kinetic and potential energy of the system. 3) Calculate the sum of virtual work done by non-conservative forces 4) Hamiltonian principle equation of the generation to obtain the differential equation of motion.

The advantage of the Hamilton principle is that inertial forces and elastic forces are not obviously used, and they are replaced by variations on kinetic energy and potential energy, respectively. Therefore, for both of these terms, only scalar processing, i.e., energy, is involved. In virtual displacement theory, the virtual work itself is scalar, but the force and displacement used to compute virtual work are all vectors. The above variational procedure differs from the virtual work procedure used previously in that the external load as well as the inertial and elastic forces are not explicitly involved; the variations of the kinetic and potential energy terms, respectively, are utilized instead. It has the advantage of dealing only with purely scalar energy quantities, whereas the forces and displacements used to represent corresponding effects in the virtual work procedure are all vectorial in character even though the work terms themselves are scalar.

2.2.4 Lagrange’s Equations The equations of motion for an MDOF system can be derived directly from Hamilton’s equation by simply expressing the total kinetic energy T, the total potential energy V, and the total virtual work Wnc in terms of a set of generalized coordinates, q1, q2, …, qN. For most mechanical or structural systems, the kinetic energy can be expressed in terms of the generalized coordinates and their first-time derivatives, and the potential energy can be expressed in terms of the generalized coordinates alone. In addition, the virtual work which is performed by the nonconservative forces as they act through the virtual displacements caused by an arbitrary set of variations in the generalized coordinates can be expressed as a linear function of those variations. In mathematical terms the above three statements are expressed in the form

T

T q1 , q2 ,..., qN ; q1 , q2 ,..., qN

(2.19)

V q1 , q2 ,..., qN

(2.20)

V

Structural Equation of Motion Establishment 31

Wnc

Q1 q1 Q2 q2 ... QN qN

(2.21)

where the coefficients Q1, Q2, … QN are the generalized forcing functions corresponding to the coordinates q1, q2, …, qN, respectively. Integrating the above equations with respect to t between t0 to t1 we get t1

T q1

t0

V q1

q1

T

q1

q2 V q2

T

q2

qN V

q2

qN

qN

T

qN

T

q1

q1

q2

Q1 q1 Q2 q2

T

q2

qN

qN

QN qN dt=0 (2.22)

Integrating the velocity-dependent terms in equation 2.22 by parts leads to t1 t0

T qi dt= qi

T qi qi

t1

t0

t1 t0

d dt

T qi

qi dt

(2.23)

The first term on the right-hand side of equation 2.23 is equal to zero for each coordinate since δqi(t0) =δqi(t1) = 0 is the basic condition imposed upon the variations. Substituting equation 2.23 into equation 2.22 gives, after rearranging terms, t1 t0

N

i 1

d dt

T qi

T qi

V qi

Qi

qi dt=0

(2.24)

Since all variations δqi (i=1; 2; …, N) are arbitrary, equation 2.24 can be satisfied in general only when the term in brackets vanishes, i.e.,

d dt

T qi

T V + =Qi qi qi

(2.25)

Equations (2.25) are the well-known Lagrange’s equations of motion, which have found widespread application in various fields of science and engineering.

32

Structural Dynamics

It can be seen that the steps for establishing the equation of motion using the Lagrange’s equations are: 1) Identify the system and figure out what objects are included in the system under consideration. Then determine the degree of freedom and choose a proper set of generalized coordinates. 2) Find the kinetic and potential energy of the system 3) Find non-conservative forces in the system 4) Substitute each of the above quantities into the Lagrange equations, the equations of motion are obtained by simplification. The above describes four basic methods for establishing the equation of motion for a dynamic system. The direct equilibration method is a simple and intuitive way to establish the equation of motion, which has been widely used. More importantly, the direct method establishes the concept of dynamic balance. The method of establishing equilibrium equations in the structural static analysis is directly extended to dynamic problems. If the structure has distributed mass and elasticity, it may be difficult to directly use the dynamic balance method to establish the equation of motion of the system. At this time, it may be more convenient to use the virtual displacement principle. It partially avoids the vector operation, the scalar operation can be used to establish the system’s equation of motion after obtaining the system virtual work. The Hamilton’s principle is another scalar method for establishing equations of motion. Based on energy method, if the work (mainly the damping force) of non-conservative forces is not considered, it is a complete scalar operation. The Lagrange’s equation used to establish the equations of motion is more common. It is the same as the Hamilton’s principle. For damping force, it is obtained through experimental tests in practice, rather than complete mathematical and mechanical analysis of structural materials and components. The mechanical damping of continuum medium is mainly caused by the medium itself, and the structural dynamics damping is different. The source for structural damping is more complex, it cannot be simply derived, but the experimental coefficient is given by actual measurement or experience.

2.3 Theory of Total Potential Energy Invariant Value of Elastic System Dynamics 2.3.1 The Main Idea of the Principle of Virtual Work In order to clarify the principle of the total potential energy invariance of the elastic system dynamics, the idea and physical concept of the principle

Structural Equation of Motion Establishment 33 of virtual work and the principle of static potential energy constant value are first introduced. The principle of virtual work is derived from the sum of the forces of the balance force on the infinitesimal virtual displacement of the system equal to zero or derived from the principle of conservation of energy. The virtual displacement is an arbitrary small displacement that people imagine to satisfy the system deformation coordination condition (constraint condition), and has nothing to do with the actual force of the system, so it is called virtual displacement. Let a particle maintain equilibrium under any force system. If there is any displacement for any reason, the total work done by this force is equal to zero. This is the virtual work theory of the particle. Virtual work, that is, one of the force system and the displacement is dummy, and each of them is independent, which is the basic concept of virtual work theory. Multiply the three equilibrium equations of elastic mechanics by the virtual displacement. After mathematical demonstration, the following virtual work equations are derived. T s

X T udv

uds v

T

dv

(2.26)

v

The left side of the equation (2.26) is δW, the sum of the external force φ and the virtual force X of the physical force. The right side is the virtual strain energy δUi of the system. Therefore, the equation (2.26) is abbreviated as

W = Ui

(2.27)

The negative value of force work W is its potential energy Ue, so, SW = -SUe it is substituted (2.27), which gives a more concise expression of the principle of virtual work.

U = (U i U e ) 0

(2.28)

Equation (2.28) is called the system total potential energy (also known as potential energy) standing value principle, where U = Ui + Ue is the total potential energy of the system and Ue is the potential energy of the external force (potential energy). When changing the total potential energy of the system, it is necessary to maintain the essence of the virtual work principle of (2.28), the external force and the stress remain unchanged,

34

Structural Dynamics

only the displacement and strain variation, let δ in equation (2.28) always maintain the effect of δ on displacement and strain variation in equation (2.26). It is not possible to express the virtual work principle (2.26) as the total potential energy standing value principle (2.28), which will result in the meaning of the variable semicolon δ deviates from the original meaning of the virtual displacement and the virtual strain in the equation (2.26), but δ is also thought as a mathematical variational symbol for total potential energy U. According to the above discussion, two points can be summarized. Virtual displacement is an arbitrarily small deformation coordination displacement in an imaginary equilibrium system, is a possible trend, independent of external forces and stresses in the system; it is not the actual displacement, and does not destroy the balance of the force system. Therefore, the external force and the internal force do not change during the virtual displacement process; Move the virtual displacement in equation (2.26) and δ before the virtual strain symbol to the integral number to obtain the principle of total potential energy standing value of the elastic system. This movement of δ in equation (2.28) does not cause changes in external force and stress of the system; so for the first-order variation of the total potential energy U of the system, only the displacement u and the strain ε vary, and the external force and the stress are not strained, so as to ensure that the equation (2.28) returns to the equation (2.26). This just reflects the requirements of the principle of virtual work.

2.3.2 Derivation of the Principle of Total Potential Energy Invariant After introducing D’Alembert’s principle and considering the damping force, the elastic system dynamics problem is transformed into the dynamic equilibrium problem. The general form of the equilibrium equation is

fs

fm

fc

Fsignu P(t ) Q

(2.29

Structural Equation of Motion Establishment 35 In the formula fs——System elastic force vector array fm = fc =

u dv——System inertial force vector array v

cu dv——System viscous damping force vector array v

F——System Coulomb Friction Vector Array P(t)——System interference force vector array Q——System gravity vector array At a given instant t, the two sides of the equation (2.29) are multiplied by the virtual displacement δu of the system, and the elastic force δuTfs is taken as the variational δUi of the virtual strain energy Ui of the system, and we can obtain

uT udv

Ui

v

uT cudv

uT Fsignu

uT P(t )

uT Q

v

(2.30) Equation (2.30) is the general expression of the principle of virtual system dynamics. It can be reduced to the following simple form

(U i Vm Vc VF VP Vg )

d

0

(2.31)

In the equation

Ui——System elastic strain energy Vm =- -uT u dv——System inertial force to do negative work v

Vc =- -uT cu dv ——System viscous damping force to do negative work v

VF =-(-uT Fsignu)——System Coulomb friction work negative value

VP = -uT P(t)——System interference force does negative work Vg = -uT Q——System gravity potential energy

36

Structural Dynamics

Because the virtual displacement is applied to the balance system, the time is instantaneously fixed, and all the forces acting on the system are unchanged. Their work is only related to the starting position and the ending position of the displacement, and is independent of the moving path of the force acting point. The definition, so they can all be seen as powerful. Thus, Vm, Vc, VF, VP, Vg are the potential energy of the system inertial force, damping force, Coulomb friction, interference force and gravity. Therefore, d

U i Vm Vc VF VP Vg

(2.32)

is the total potential energy of the elastic system dynamics, and it is also the general calculation formula of the system. Imitating the equation (2.28), we call the equation (2.31) as the principle of the total potential energy of the elastic system dynamics. The right subscript of the variable semicolon in the equation also emphasizes the total potential energy Пd of the elastic system dynamics. In the case of variation, it is necessary to maintain the essence of the virtual work principle of (2.31), that is, only for elastic strain and displacement variation, inertial force, damping force, interference force, gravity and other force components are invariant, that is, in calculation, Vm, Vc, VF, VP, Vg, all strains and displacements are considered variables, and the corresponding forces are specified. The physical meaning of equation (2.31) is: after introducing D’Alembert’s principle, at a given instant t, the first-order variation of the total potential energy Пd of the linear system dynamics must be equal to zero, and Пd takes a constant value. The general formula hold value is based on the functional variational principle. Here, the total potential energy Пd is constant according to the principle of virtual work. Obviously there is no reason to calculate the first-order variation of X by the variational method. The strain and displacement components in Пd should only be changed according to the physical concept of the virtual work principle. In summary, the basic steps of establishing the equation of motion by the principle of the total potential energy of the elastic system are as follows: Firstly, the potential energy of each force acting on the system is derived, and the total potential energy of the motion system is obtained Пd; then, the vibration differential equation of the system is derived from the condition of δε Пd = 0. The greatest advantage of the principle of the total potential energy of the elastic system dynamics is that it is not necessary to distinguish between forces and non-potentials in the process of system

Structural Equation of Motion Establishment 37 dynamics modeling, and consider whether the external potential field is stationary and whether the constraints are stable.

2.4 Influence of Gravitational Forces Consider now the system shown in Figure 2.7. In this case, the system of forces acting in the direction of the displacement degree of freedom is that set shown. Using Equation (2.4), the equilibrium of these forces is given by

mu t

cu t

k u t

ust

P t

W

(2.33)

where W is the weight of the body taken as kust = W, then we have,

mu t

ku t =P t

cu t

(2.34)

Comparison of equations (2.4) and (2.33) demonstrates that the equation of motion expressed with reference to the static equilibrium position of the dynamic system is not affected by gravity forces. For this reason, displacements in all future discussions will be referenced from the static equilibrium position and will be denoted u(t) (i.e., without the overbar); the displacements which are determined will represent dynamic response. Therefore, total deflections, stresses, etc., are obtained by adding the corresponding static quantities to the results of the dynamic analysis.

k

c

c

k

O

c

k O’

m

FI

FS

FD

O’

m m

m

u w

u

u w P(t)

Figure 2.7 Influence of gravity on SDOF equilibrium.

w P(t)

38

Structural Dynamics

2.5 Influence of Support Excitation Dynamic stresses and deflections can be induced in a structure not only by a time-varying applied load, but also by motions of its support points. Important examples of such excitation are the motions of a building foundation caused by an earthquake or the base support of a piece of equipment due to vibrations of the building in which it is housed. The horizontal girder in this frame is assumed to be rigid and includes all the moving mass of the structure. The vertical columns are assumed to be weightless and inextensible in the vertical (axial) direction, and the resistance to girder displacement provided by each column is represented by its spring constant k/2. The mass thus has a single degree of freedom, u(t), which is associated with column flexure; the damper c provides a velocity proportional resistance to the motion in this coordinate. The equilibrium of forces for this system can be written as

FI t

FD t

FS t

0

(2.35)

in which the damping, elastic and inertial forces can be expressed as,

FI t

mu0 t

FD t

cu t

FS t

ku t

m ug t

u t (2.36)

where u0(t) represents the total displacement of the mass from the fixed reference axis. Substituting for the inertial, damping, and elastic forces in Equation (2.36) yields

mu0 t

cu t

ku t

0

(2.37)

Before this equation can be solved, all forces must be expressed in terms of a single variable, which can be accomplished by noting that the total motion of the mass can be expressed as the sum of the ground motion and that due to column distortion, i.e., Expressing the inertial force in terms of the two acceleration components obtained by double differentiation of equation (2.37) and substituting the result into equation (2.36) yields

Structural Equation of Motion Establishment 39

mug t

mu t

cu t

ku t

0

(2.38)

or, since the ground acceleration represents the specified dynamic input to the structure, the same equation of motion can more conveniently be written

mu t

cu t

ku t

mug t

Peff t

(2.39)

In this equation, Peff (t) denotes the effective support excitation loading; in other words, the structural deformations caused by ground acceleration üg(t) are exactly the same as those which would be produced by an external load Peff(t) equal to –müg(t). The negative sign in this effective load definition indicates that the effective force opposes the sense of the ground acceleration. In practice this has little significance inasmuch as the engineer is usually only interested in the maximum absolute value of u(t); in this case, the minus sign can be removed from the effective loading term. An alternative form of the equation of motion can be obtained by using equation (2.37) and expressing equation (2.36) in terms of u0(t) and its derivatives, rather than in terms of u(t) and its derivatives, giving

mu0 t

cu0 t

ku0 t

cug t +kug t

(2.40)

In this formulation, the effective loading shown on the right side of the equation depends on the velocity and displacement of the earthquake motion, and the response obtained by solving the equation is total displacement of the mass from a fixed reference rather than displacement relative to the moving base. Solutions are seldom obtained in this manner, however, because the earthquake motion generally is measured in terms of accelerations and the seismic record would have to be integrated once and twice to evaluate the effective loading contributions due to the velocity and displacement of the ground.

Exercises 2.1 Try to answer what are the basic elements for structural dynamic analysis and find out the mechanism and mode of damping. 2.2 What is D’Alembert’s principle? Explain how the principle is employed in vibration problems. 2.3 Write the equation of motion for the inverted pendulum using virtual displacement method.

40

Structural Dynamics

Figure 2.8

References 1. Clough, R.W. and Penzien, J., Dynamics of Structures, Computers and Structures, 2005. 2. Wilson, J.F., Dynamics of Offshore Structures, Second Edition, John Wiley & Sons, Inc, 2003. 3. Chopra Anil, K., Dynamics of Structures, Prentice Hall, 2000. 4. Timoshenko, S., Vibration Problems in Engineering, Van Nostrand, 1975. 5. Paz, M., Structural Dynamics-Theory and Computation, Van Nootrand Company.Inc, 1985. 6. Hibbeler, A.C., Structural Analysis, Prentice Hall, 2011. 7. Lanczos, C., The Variational Principles of Mechanics, Dover Publications Inc, 1986. 8. Worden, K. and Manson, G., Random vibrations of a multi-degree-of-freedom non-linear system using the volterra series. J. Sound Vib., 1999. 9. Hsueh, W.J., Vibration transmissibility of a unidirectional multi-degree-offreedom system with multiple dynamic absorbers. J. Sound Vib., 2000. 10. De Wachter, S. and Tzavalis, E., Detection of Structural Breaks in Linear Dynamic Panel Data Models, Computational Statistics and Data Analysis, 2012. 11. Cluni, F., Gioffrè, M., Gusella, V., Dynamic response of tall buildings to wind loads by reduced order equivalent shear-beam models. J. Wind Eng. Ind. Aerodyn., 2013. 12. Zhao-Dong, X. and Le-Wei, M., Dynamics of Structures, Science Press, Beijing, 2007.

3 Single Degree of Freedom Systems The concept of dynamic degrees of freedom has been introduced in Chapter 2, in which the system with one dynamic degree is called Single Degree of Freedom System (SDOF). The SDOF system is the simplest dynamic system. It is of great significance to study the dynamic characteristics and vibration problems of Single Degree of Freedom Systems for the following aspects: (1) SDOF system has the basic characteristics of a general dynamic system, including all the physical quantities and basic concepts involved in the dynamic analysis of structure, which is the basis of learning structural dynamics. (2) Many practical dynamic problems can be analyzed and calculated directly as a SDOF system, such as one-storied factory and water tower, etc. Figure 3.1 presents several mechanical models commonly used in the structural dynamic analysis as a SDOF system. The following picture shows how to simplify complex structure into a SDOF. This chapter deals with the SDOF systems’ dynamic characteristics as well as its responses under free vibrations at the beginning, then the solving process of dynamic response under the action of harmonic, periodic, impulsive, and random loading will be introduced. Finally, this chapter will end with the discussion of damping, vibration energy, vibration measurement, and the principle of vibration isolation.

3.1 Response of Free Vibrations Chapter 2 has discussed how to establish the motion differential equation of dynamic systems. As is shown in Figure 3.2 (as example 2.1), the equation of motion for this SDOF system is found to be

mu(t ) cu(t ) ku(t ) P(t )

(3.1)

Yong Bai and Zhao-dong Xu (eds.) Structural Dynamics, (41–125) © 2019 Scrivener Publishing LLC

41

42

Structural Dynamics

m

k k

k c

(a) Single-story factory

(c) Water tower structure

(b) Mass-spring-damping system

(d) Simplification model of single-story structure

(e) Cantilever elastic support structure

(f) Simply supported beam structure

Figure 3.1 Commonly used single degree of freedom systems.

m

Simplify Machine k Ground

Figure 3.2 To simplify a complex structure.

c

Single Degree of Freedom Systems 43 Structural free vibration is a vibrational process in which the structure leaves the equilibrium position when disturbed and no longer affected by any external force, i.e., P(t)=0, and the equation of motion will be

mu(t ) cu(t ) ku(t ) 0

(3.2)

When the dynamical system has the damping term, which means the system has an energy dissipation mechanism, the reaction of the dynamic system will be influenced directly. The SDOF system without damping is called the undamped SDOF system, while the one that has damping is termed the damped SDOF system. Free vibration of the undamped SDOF system will be introduced first.

3.1.1 Undamped Free Vibrations For the undamped SDOF system, c = 0, according to the Eq. (3.2), the motion equation of the structure can be expressed as

mu ku 0

(3.3)

Assuming the solution is in the following form

u(t ) Ae st

(3.4)

Where s is undetermined constant and A is constant coefficient. Substituting Eq. (3.4) into Eq. (3.3) leads to

(ms 2 k )Ae st

0

(3.5)

It is evident that the two eigenvalues of Eq. (3.5) are s1 = i n and s2 = −i n. k , which is a constant parameter related to Adopting the notation of n2 the properties of the structure. m Utilizing the relationship between exponential function and trigonometric function of eix = cos x + i sin x and e−ix = cos x – i sin x, solutions of Eq. (3.4) may be expressed by sine function and cosine function in the following form

u(t ) A cos

n

t B sin

n

t

(3.6)

44

Structural Dynamics

In this formula, A and B are two new undetermined constants which will be confirmed by the initial conditions of the system of A = u (0). If the initial conditions are defined at t = 0, u (t) = u (0), the solution will be concluded, which is given according to Eq. (3.6)

u(t )

n

A sin

n

t

n

B cos

n

t

(3.7)

So that leads to

A u(0), B u(0)

(3.8)

n

Introducing Eq. (3.8) into Eq. (3.6) leads to the solution of the free vibration of undamped system:

u(t ) u(0)cos

n

u(0)

t

sin

n

t

(3.9)

n

Obviously, the undamped free vibration of the system is a simple harmonic motion. The structural dynamic characteristics can usually be represented by three physical quantities of n, Tn , and fn in its free vibration analysis. is the natural vibration circular frequency, which can be expressed as: n

k m

n

(3.10)

Tn is the time required to complete a cycle of structural motion, which is called the natural vibration period of the structure, which can be expressed as:

Tn

2

2 n

m k

(3.11)

fn is called as structural natural vibration frequency, namely the number of cycles of vibration per unit time, whose unit is Hz (times per second). It is commonly employed as a measurement of the speed of structural vibra1 , i.e., tion which is defined fn Tn

Single Degree of Freedom Systems 45 n

fn

(3.12)

2

It can be seen that the natural vibration circular frequency n is only related to the stiffness, k, and quality, m, while the natural vibration frequency, fn, and the natural vibration period, Tn, are only related to n. Thus, , f , and Tn are all the structural inherent characteristics, which are only n n related to the structure itself. They are all important physical quantities that can reflect structural characteristics. Eq. (3.9) is plotted in Figure 3.3. It demonstrates the curve of the undamped free vibration response changing with time. The curve’s value is equal to the initial displacement, u(0), and the slope of the curve goes to the initial velocity, u(0), at the initial moment (t = 0). The curve changes along the slope’s direction, and it will reach its maximum value u0 after a period of time, namely

u0

max u(t )

u(0)

u(0)

2

2

(3.13)

n

Where u0 is the amplitude of free vibration system. In line with the trigonometric knowledge of higher mathematics, the response of undamped free vibration of the Eq. (3.9) can be expressed in the following form:

u(t ) u0 cos( Phase angle

n

t

)

(3.14)

is given by the following formula

tan

1

u(0) nu(0)

(3.15)

The phase angle indicates that the motion of the dynamic system lags behind the angular distance of the cosine, as shown in Figure 3.4. k

m P(t)

c

Figure 3.3 Single degree of freedom system.

46

Structural Dynamics Tn = 2π/ωn

.

u(0)

u(0)

u0

u(t)

t /ω

Figure 3.4 The response of undamped free vibration system.

What can be seen from Figure 3.4 is that the displacement response of the undamped SODF system is a simple harmonic motion centered on a balanced position, which is connected with displacement u0, initial velocity u(0) , and natural vibration period Tn.

3.1.2 Damped Free Vibrations If damping exists in the dynamic system, its free vibration equation will be mu(t ) cu(t ) ku(t ) 0. Commanding u(t) = Aest and substituting it into the equation leads to

c 2m

s1,2

c 2m

2 2 n

(3.16)

k is the natural vibration circular frem quency of undamped system which has been defined previously. This expression can describe three different types of motions depending on the signs of the radical above, which can be positive, negative, or zero. When the stiffness and mass of the structural system is determined, the value of the radical entirely depends on the damping coefficient, c. The value of the radical may be greater than zero when c is larger, which will be opposite when c is much smaller. These two conditions correspond to two completely different kinds of motion state, respectively. When the radical is positive, which results in two real solutions s1 and s2, the system will not undergo reciprocating oscillation. But if the radical is less than zero, these two solutions will be of different pluralities, which will give the system In the formula above,

n

Single Degree of Freedom Systems 47 a motion of reciprocating vibration. There is also a possibility in which the radical equals 0, which is the demarcation line for these two motion states described above; the damping in this state is called Critical Damping. 1. Critical Damping To order the radical in Eq. (3.16) is zero leads to the value of critically damped ccr

ccr

2m

n

2 km

(3.17)

So the critical damping ccr is also a constant completely determined by the stiffness and mass of the structure. The two characteristic roots of the Eq. (3.16) will be:

s1

s2

(3.18)

n

Therefore, the solution of the equation will be:

u(t ) (C1 C2t )e

nt

(3.19)

In the formula, C1 and C2 are the undetermined coefficients which are confirmed by the initial conditions. Supposing the initial displacement and the initial velocity are u (0) and u(0) respectively, then C1 = u(0), C2 u(0) nu(0), so the final form of the critical damping system will be

u(t )

u(0)(1

n

t ) u(0)t e

nt

(3.20)

What can be found out is that the critical damping system will not oscillate near the static equilibrium position. The displacement amplitude of the damped system oscillates with decreasing exponentially up to zero over time, where the system will not oscillate anymore and return to the static equilibrium position. The so-called critical damping is the minimum value of damping in the case of not oscillating in the free vibration responses. s1 and s2 will be different negative real roots when the damping of structural system is greater than the critical damping, in which the displacement amplitude of motion equation will also decay exponentially. In this

48

Structural Dynamics

situation, only if the damping of the system is less than the critical damping will the system vibrate freely. 2. Damping ratio The damping ratio, ζ, is often used in theoretical calculation and engineering applications to indicate the size of structural damping, and ζ is the ratio of damping coefficient c to critical damping ccr , i.e.

c ccr

c 2m

(3.21) n

(1) When c = ccr, ζ = 1, the damping is namely the so-called critical damping mentioned above (2) When c < ccr, ζ < 1, the damping is underdamped, and the structural system is known as an underdamped system (3) When c > ccr, ζ > 1, the damping is overdamped, and the structural system is known as an overdamped system. The damping ratio ζ is generally determined by experiment, which will be described in the later chapter. The damping coefficient c = 2m nζ of dynamic analytic calculations can be given by the damping ratio. Similarly, the damping ratio is a characteristic parameter to measure energy dissipation capability of dynamic system. According to statistics, the damping ratio ζ is 0.02 for steel structure, and 0.05 for reinforced concrete structures, while the damping ratio ζ of the energy dissipation system or the dynamic system added damper (shock absorber) described in Chapter 7 will be 0.10 ~ 0.20 generally. 3. Underdamped systems Underdamped system indicates that the structural damping is less than the critical damping, i.e. c < ccr, ζ < 1, substituting c = 2m nζ into Eq. (3.16) leads to two characteristic roots:

s1,2 2

2 n

n

1

(3.22)

1 will be a complex number which may leads the two characteristic roots are both complex numbers. Adopting the same analysis method

Single Degree of Freedom Systems 49 used in undamped free vibration, the solution of underdamped free vibration system satisfying initial conditions can be obtained.

u(t ) e

nt

u(0)cos

D

u(0)

t

n

u(0)

sin

D

t

(3.23)

D

Where D is the natural circular frequency of vibration of the system 2 . where D n 1 By comparing the responses of free vibration between the undamped system (Eq. (3.9)) and the damped system, the characteristic of reciprocating near the equilibrium position by circular frequency, n, and the constant amplitude of motion can be understood when free vibration occurs in the undamped system. However, the free vibration of the underdamped system has an invariable circular frequency D while oscillating near the center position, but its natural vibration frequency will gradually become smaller because of the existence of damping. Meanwhile, Eq. (3.23) indicates that free vibration of the damping system is a vibration process of amplitude attenuation, in which the displacement amplitude decays in the regular of exponential function e nt over time. It is also shown that the damping force constantly consumes energy and weakens the vibration until it disappears. When ζ = 0, Eq. (3.23) will degenerate into the solution in the form of undamped free vibration as the Eq. (3.9). The damping ratio ζ is relatively smaller in the actual engineering structure, and the maximum damping ratio is no more than 0.20. at this time, 0.98 n D n 1 0.04 n . Therefore, the influence of damping on the circular frequency will generally be ignored in the dynamic calculation for a practical engineering structure, namely D = n. In order to clearly show the effect of the damping ratio ζ when taking different values on the free vibration motion, the free vibration curve is shown in Figure 3.5. 4. Overdamped systems Overdamped system means damping of the structure is greater than the critical damping, namely c > ccr, ζ > 1. In this case, these two characteristic roots will be two negative real numbers

s1,2

2 n

n

1

n

D

(3.24)

50

Structural Dynamics u (t)

u (0)

ζ=1

ζ = 0.02

ζ=0

ζ>1 t

ζ = 0.05

ζ = 0.1

Figure 3.5 Influence of different damping on free vibration motion.

2 1 is the natural frequency of damped vibration of Where D n overdamped system when considering damping which gives:

u(t ) e

nt

(C1 sinh

D

t C2 cosh

D

t)

(3.25)

The constants C1 and C2, in the formula are determined by the initial conditions. Substituting them in, Eq. (3.25) gives:

u(t ) e

nt

(

u(0)

n

u(0)

sinh

D

t u(0)cosh

D

t)

(3.26)

D

In this case, the system movement will be a periodic motion, which is attenuated by the exponential law, due to the damping being too large, as shown in Figure 3.5. Example 3.1 Figure 3.6 shows a calculation diagram of a single-story building. Assuming the beam stiffness is infinite, EI = , the weight of roof system, beam, and part of columns can be concentrated in the beam, which is m = 1×104 kg n order to determine the dynamic characteristics of the gantry in the situation vibrating horizontally, following vibration

Single Degree of Freedom Systems 51 experiments were carried out: Applying a horizontal force, P = 98kN makes the gantry have a displacement of u0 = 0.5cm then the structure will have a free vibration. If the period T = 1.5s and the lateral displacement of the beam is measured to be u1 = 0.4cm after one cycle. Calculate the damping coefficient of the gantry and the amplitude after 5 cycles of vibration.

u0 m

3m

k

c

k

Figure 3.6 Influence of different damping on free vibration motion.

Solution: according to the subject

k

P u0

98 103 0.5 10 2

1.96 107 N / m 1.96 104 kN / m

So

n

k m

1.96 107 104

44.2719 rad / s

The damping ratio

1 u0 ln 2 u1

1 0.5 ln 2 0.4

0.0355

52

Structural Dynamics The damping coefficient

c 2m

n

2 104 44.2719 0.0355 3.1433 104 ( N s ) / m u5

u0 e

10

0.5 e10

0.16 cm

3.1.3 Damping and Its Measurement 1. Damping form The source and brief introduction of damping commonly used have been described in Chapter 2. Damping generally comes from the internal friction when material deforms, the friction in the connection of structure and the external media around the structure. According to the energy dissipation mechanism, three main different damping theories were proposed. (1) Viscous damping Viscous damping considers the magnitude of the damping force to be proportional to the velocity, while its direction is opposite to the speed which can be expressed as:

FD (t ) cu(t )

(3.27)

Where the subscript D in FD(t) represents damping; FD(t) is the damping force; c is the damping coefficient; u(t ) is the velocity of the particle. The motion equation which established by the theory of viscous damping is easy to solve, so this theory is widely used in current dynamic analyses. (2) Complex damping Complex damping is also called structural damping, hysteresis damping, or material damping. This damping theory states that the damping force is proportional to displacement in the simple harmonic vibration, and its phase is the same with speed, which means the phase of damping force is ahead of displacement by 90°or the time ahead of the quarter cycle. The damping force can be expressed in the following complex number form

FD (t ) i ku(t )

(3.28)

Single Degree of Freedom Systems 53 In the formula, is the complex damping coefficient, k is the stiffness coefficient. Complex damping can better reflect the energy dissipation mechanism of the friction in the material, which has wide applications. (3) Friction damping Friction damping holds that the damping force is a constant, which is equal to the friction, and its direction is opposite to the velocity. The friction damping, also known as dry friction damping, can be expressed as:

FD (t )

N

u(t ) u(t )

(3.29)

Where is the coefficient of friction, N is the positive pressure between the friction of the contact surface. Friction damping is generally applied to the dynamic system where the frictional resistance is dominant, while coulomb damping is more suitable in the engineering structures installing friction dampers or the friction-type isolation structures. Furthermore, there are other damping models such as fluid damping, exponential damping, etc., and interested readers can refer to the relevant books. 2. Motion attenuation and measurements of damping ratio What can be seen from Figure 3.5 is the structural free vibration response is greatly affected by damping (damping ratio), attenuation of the free vibration response of dynamic systems with larger damping ratio is faster than that of dynamic systems with smaller damping ratio. According to this principle, we can obtain the damping ratio ζ of the structure through the structural free vibration test, as shown in Figure 3.7. If there are two amplitude values of un and un+1 which are separated by one cycle in the damped free vibration system, the simplest way to measure the damping ratio ζ is to calculate the ratio of the two values. The Eq. (3.23) can be expressed as:

u(t ) Ce un un 1

Ce Ce

(tn Δt )

nt

sin(

t

)

e

Δt

D

tn

sin( D tn ) sin D (tn Δtt )

(3.30)

sin( sin

t ) Δt ) D (tn D n

(3.31)

54

Structural Dynamics u Ae–ξωτ u(0)

u(0)

ui ui+1 t

O

Td

ti

Figure 3.7 Free vibration decay curve.

There is

D

t = 2 . so

un un 1

1 e

e

Δt

Δt

(3.32)

It can be seen that the ratio of adjacent vibration peak value is only related to the damping ratio and has nothing to do with the value of n. Taking the logarithm on both sides of the Eq. (3.32), we will get the logarithmic decay rate:

ln

2

ui ui 1

1

2

(3.33)

For a small damping system, the value of ζ is very small, so approximately (1 – ζ 2) 1. The approximate formula of the damping ratio of small damping system obtained from Eq. (3.33) will be:

2

(3.34)

Using the above formula, the damping ratio ζ can be calculated from the two measured adjacent amplitude values. However, the free vibration

Single Degree of Freedom Systems 55 will decay slowly when the damping of the structure is minute, then using a vibration peak ratio of several cycles apart to calculate the damping ratio of the structure will get higher accuracy at this time. The ratio of these two amplitudes which separated by m cycles is

un un m

un un un 1 un

un m 1 un m

1 2

e

mΔt

(3.35)

Thus, the logarithmic attenuation rate can be obtained

ln

un un m

2m

(3.36) D

D

(3.37)

2m

In the experiment, the damping ratio is sometimes calculated by the number of times J50%, when the vibration peak decays to 50%, and at this 1 ln 2, time, J 50%

1 ln 2 2 J 50%

0.11 J 50%

(3.38)

When the damping ratio of the system is calculated by the method mentioned above, there will only be information of the two peak points on the system vibration decay curve being used. There is another way to obtain the exponential decay law of the vibration amplitude by fitting the peak point decay curve, then the damping ratio which has an important effect on vibration attenuation can be calculated. Interested readers can refer to the relevant books on this subject. The following sections of this chapter will also introduce other methods of measuring damping ratios. Example 3.2 Structural parameters are the same as example 3.1. If using a jack to make m have a lateral displacement of 20mm, and then suddenly releasing it, the system will vibrate freely. The lateral displacement was measured to be 10 mm after 4 cycles of vibration. Please determine: (1) the damping ratio and damping coefficient of the structure (2) the amplitude after 10 cycles of vibration.

56

Structural Dynamics

Solution: As can be seen from example 3.1

k 1.96 104 kN / m So the natural vibration period T

the circular frequency

2 T

n

m k

2

2

m 1.96 107

0.0014 m

4.4880 103

m (1) Damping ratio and damping coefficient The damping ratio is known from Eq. (3.39)

1 u ln 0 4 2 u4 c 2m

n

2m

1 20 ln 8 10

4.4880 103

0.0276

0.0276 247.7376 m

m

24773.76

(2) As can be seen from Eq. (3.39)

u10

e

u0

20

20

20

e

3.531 mm

Therefore, the amplitude after 10 cycles of vibration is 3.531 mm. 3. Free vibration test The damping ratio ζ of an actual structure cannot be determined by the theoretical analysis method since this measurement could produce a large error. It must be determined by the free vibration test. The free vibration test provides a way to determine the structural damping ratio. According to the above derivation, when the structural damping is small (damping ratio 1, (R d )max . As the (2 1 2 ) damping ratio increases, the peak of the dynamic amplification coefficient tends to decrease. The maximum 1 2 2, value corresponding to the frequency is (1) When

n

indicating that the maximum dynamic response of the damping system will not occur at ω = ωn, but take place at n

1 2 2. / n = 1, R d

1 , it is always considered that the 2 maximum reaction (resonance) occurs approximately at  = n in practical engineering. 2 , Rd ≤ 1, any ζ is tenable. In some (4) When / n dynamic system designs, if the system is designed rationally, the frequency ratio can be made within this range, so that the dynamic response is effectively reduced.

(3) When

Example 3.4 The basic dynamic system shown in Figure 3.14 has the following characteristics: m = 10kg, k = 104 N / m. If the system is subjected to a resonant harmonic load ( = n) and the initial state is stationary, please set the value of the dynamic amplification coefficient Rd(t) after five cycles ( t = 10 ), (1) no damping system; (2) ζ = 0.02; (3) ζ = 0.05 O

u(t)

k m c

Figure 3.14 Simple basic dynamic system.

P(t)

68

Structural Dynamics

Solution: According to the given information:

m 10 kg k 104 KN / m u(0) u(0) 0

n

104 10

k m

31.6228 rad / s

(1) c = 0, namely no damping

Rd (t )

1 (sin t 2

t cos t )

1 sin(10 ) 10 cos(10 ) 2

5

15.7080

1 cos10

11.6628

(2) ζ = 0.02

Rd (t )

1 e 2

t

1 cos t

1 e 2 0.02

0.02 10

(3) ζ = 0.05

Rd (t )

1 e 2

t

1 e 2 0.05

1 cos t 0.05 10

1 cos10

7.9212

3.2.4 Resonance Reaction 2 It has been said that when , the peak value of dynamic n 1 2 amplification coefficient is the maximum, for the occurring of resonance. This is obtained by solving the first derivative of the dynamic amplification coefficient Eq. (3.62), which is equal to zero.

Single Degree of Freedom Systems 69 1 1 , Rd 1, it can be assumed and the value of Rd 2 n 1 . This suggests that Rd is a large value in is maximum, i.e., ( Rd )max 2 the case of small damping with , which increases the amplitude n When ζ2

dramatically, so the interval of 0.75

1.25 is called the resonance n

region. When the structure resonates,

=

n

, from Eq. (3.59) can be obtained:

ust 2

C 0, D

(3.67)

Substituting Eq. (3.67) into Eq. (3.60) and then command it to satisfy zero initial conditions can lead to:

1 ust , B 2

A

1 2

2 1

ust

(3.68)

Finally, the resonance reaction satisfying the zero initial condition is obtained:

u(t )

ust e 2

nt

(cos

D

t

2

1

sin

D

t ) cos

n

t

(3.69)

When the damping is small, D = n, the sine term in Eq. (3.67) has little effect, and at this point, the Eq. (3.69) becomes:

u(t )

ust (e 2

nt

1)cos

n

t

(3.70)

When ζ = 0, the resonance reaction with zero initial condition can be obtained by the Eq. (3.70)

u(t )

ust ( 2

n

t cos

n

t sin

n

t)

(3.71)

70

Structural Dynamics

3.2.5 Solution of Damping Ratio In the actual structure, the damping and energy dissipation mechanism has not been fully understood, so the damping coefficient of many structures must be obtained from the experimental method directly. There are usually three methods to calculate the structural damping ratio by experiments: logarithmic decay rate method, bandwidth method, and resonance amplification method. (1) Logarithmic Decay Rate Method This method has been introduced in the first section of this chapter in the free vibration of SDOF systems, namely using the logarithmic decay rate method to calculate the viscous damping ratio of the SDOF systems. The damping ratio of the system can be obtained by the logarithm value of the free vibration amplitude ratio (usually take the amplitude which has the interval of m cycles)

2 m

2 m

(3.72)

D

un . The reason why the damping ratio ζ can be obtained un m by the free vibration method is that the speed of the free vibration decay is controlled by ζ, or the attenuation law of free vibration, which can clearly reflect the influence of the damping ratio ζ. The main advantage of this free vibration method is that it requires the least amount of instruments and equipment, while any simple and convenient method can be used to get the vibration and the required measurement is only the relative displacement amplitude. Where

ln

(2) Bandwidth Method Bandwidth method, also known as half-power point method, is based on the characteristics of dynamic amplification coefficient Rd. From the amplitude-frequency characteristic curve of the dynamic amplification coefficient Rd, it can be seen that the Rd curve shape is completely controlled by the damping ratio, and when the damping ratio is large, Rd is fat (wide); when ζ is small, Rd is thin (narrow). The damping ratio of the system can be determined by the width of the half power point. As the amplitude-frequency characteristic curve shown in Figure 3.15. and b are the two frequency points corresponding to the points of Rd a whose amplitude is equal to 2 / 2 times the maximum amplitude, which is called the half power point. When the damping is relatively small, the

Amplitude u

Single Degree of Freedom Systems 71

a

√2

umax

umax

b

ust

0

ωn

ωn

Frequency ω

ω = ωn

Figure 3.15 Calculating damping ratio on amplitude-frequency characteristic curve.

relationship between the half power point and the damping ratio ζ is as follows b

a

(3.73)

b

a

(3.74)

b

a

2

n

Or

The damping ratio in the above formula is calculated by the circular frequency, and may also use the engineering frequency to calculate the damping ratio, i.e.,

fb f a 2 fn

(3.75)

Where fn is the engineering natural vibration frequency of the corresponding undamped dynamic system. 1 Notice ( R d )max and the amplitude is equal to 2 2 1 2 ( Rd )max . 2

72

Structural Dynamics The corresponding frequency satisfies the following equation

1 1 ( /

n

)2

1

2

2 ( /

n

)

1

22

2

1

(3.76)

2

Taking the reciprocal of Eq. (3.76) on both sides and squaring it results in: 4

2 2

1 8 2 (1

2(1 2 ) n

2

) 0

n

(3.77)

Eq. (3.77) is a quadratic equation about ( / roots

n

)2, which can have two

2

(1 2 2 ) 2

1

2

(3.78)

n

Eq. (3.78) corresponds to the larger root b when it takes a positive sign, otherwise corresponds to the smaller root a. The damping of the general engineering structure is relatively small, and the square term of the ζ in Eq. (3.78) can be neglected, so:

1 2

1

(3.79)

n

The two roots corresponding to the half power point are: b

1

a

,

n

1

(3.80)

n

The relation between the half power point frequency, damping ratio ζ is obtained by Eq. (3.80) b

a n

2

b

and

, and the

a

(3.81)

Single Degree of Freedom Systems 73 Thus, Eq. (3.73) is obtained. If substituting the relationship

b

a

2

n

from Eq. (3.80) into Eq. (3.73), Eq. (3.74) which used to calculate n is also obtained. Bandwidth method (half power point method) uses forced vibration tests, which can be used for not only SDOF system but also MDOF system. While Making the resonant frequency sparse is required for the MDOF system. The multiple natural vibration frequencies should be far apart, and it must not be affected by adjacent frequencies when determining the half power point corresponding to a natural vibration frequency. By using this method, the required static response in the resonant amplification method can be avoided. However, it is necessary to accurately graph the half power range and the reaction curve at resonance. (3) Resonance Amplification Method Similarly, starting from the dynamic amplification coefficient, the amplification coefficient of arbitrary frequency is the ratio of the response amplitude of the frequency to the zero-frequency (static) response amplitude. From the previous analysis it has been shown that the damping ratio is closely related to the resonance amplification coefficient, and 1 Rd . When resonance (ω/ωn = 1) 2 2 2 1 ( / n) 2 ( / n) u0 u0 ( n ) 1 , obviously, once the dynamic ( occurs, Rd ( n ) n) 2 ust ust amplification coefficient Rd curve is obtained, the damping ratio can be determined

1 2 Rd (

n

)

ust 2u0 ( n )

(3.82)

Since it is not easy to determine u0( n) from the dynamic amplification coefficient curve, the value u0m is generally used instead, u0m = max(u0), then

ust 2u0m

(3.83)

When the damping ratio is relatively small (e.g., ζ n, it can be realized by reducing the natural vibration frequency n of the instrument. In practice, it can be realized by reducing the spring stiffness k or increasing the mass m. Therefore, inertial displacement sensors are generally more flexible. From the above analysis we can see that both acceleration sensors and inertial displacement sensors work in a certain frequency band. 4. Speed Sensor The common speed sensor is inertial speed sensor, the working principle is that when the sensor is vibrating with the structure or the top rod of the sensor, while moving coil is connected with the structure. Because of the vibration of the structure, the moving coil of the sensor moves in the magnetic field, the induced magnetic field is generated by cutting the magnetic lines now. The magnitude of the induced electromotive force is directly proportional to the moving speed of the moving coil; therefore, the vibration velocity of the structure can be determined by measuring the induction electromotive force.

114

Structural Dynamics

3.7.4 Data Acquisition and Analysis System The traditional method of vibration measurement is to change the continuously changing vibration into continuous voltage signals for display and recording. The common instruments are pen recorders, ray oscilloscopes, tape recorders, and so on. The continuous changing signals are called analog signal. The disadvantage of analog signal is the low precision of its display or recording, poor anti-interference ability, and it is inconvenient for further analysis and processing. Dynamic data acquisition is used to convert analog signals into digital signals. Digital signals are easy to store, transmit, and analyze, so the signal conversion device is needed in the acquisition system. More information about electronics and cybernetics is presented here; readers who are interested in the data collection and analysis system can refer to the relevant books.

3.8 Vibration Isolation Principle The building or machine’s vibration which is beyond its allowable range affects its normal operation and service life. It is also a form of vibration pollution, affecting the normal work of the surrounding equipment as well as human health. Isolating vibration effectively is very important. Vibration isolation in engineering is divided into two cases: (1) Active vibration isolation: Prevent vibration output. For example, the operating device can produce oscillation force, and these forces may produce harmful vibration in the support structure. (2) Passive isolation: Prevent vibration input. For example, isolation design in structural seismic problems, vibration isolation of precision instruments and equipment installed on vibrating structures or foundations.

3.8.1 Active Vibration Isolation The vibration source is a machine or a device, and the purpose of active vibration isolation is to reduce the force transmitted to the supporting structure or foundation. In fact, it is the separation of forces, and its mechanical model is shown in Figure 3.28.

Single Degree of Freedom Systems 115

P(t) = P0 sin ωt

k

c

Figure 3.28 Single degree of freedom active vibration isolation system.

It is assumed that the unbalanced force produced by the machine is P0 sin t. is the rotation rate of the machine (angular speed); m is machine mass (set as rigid mass); k and c are the total stiffness and damping of the isolator; PT is the force that is transmitted from the isolator to the foundation. The machine is mounted on a SDOF system, as shown in Figure 3.28. The force transmitted through the isolator to the foundation is:

PT

FS

FD

ku cu

(3.194)

This is a simple harmonic vibration problem, and its solution is u(t) = ustRd sin( t – ). At the same time, the velocity relative to the basement is u(t ) ust Rd cos( t ), and by substituting them to (3.194), the force on the ground can be obtained:

PT (t ) ust Rd k sin( t

) c cos( t

)

(3.195)

Then the amplitude of the action force PT is

PT max substitute ust

ust Rd k 2 c 2

2

(3.196)

P0 , c = 2m nζ into formula (3.196), then: k

PT max

P0 Rd 1 (2

/

n

)2

(3.197)

116

Structural Dynamics

We call the ratio of the maximum of the force act on the foundation PTmax and the magnitude of the force on the system P0 force transmissibility, which we can denote by TP:

TP

PT max P0

1 1 ( /

2 ( / n

)2

2

n

)

2

2 ( /

n

)

2

(3.198)

TP is the amount that reflects the positive vibration reduction effect when 2, TP < 1. That means improving the frequency ratio of vibration ison

lation system ( / n) can achieve vibration isolation. Therefore, in order to achieve the purpose of vibration isolation, the method of reducing the frequency of the natural vibration circle n can be adopted, that is, the vibration isolation effect can be improved by reducing the stiffness of vibration isolation element or increasing the mass of the instrument. Of course, reducing the stiffness of the isolator or increasing the quality of the instrument will lead to an increase in the static displacement of the instrument or equipment. The actual vibration reduction design should be selected between the smallest possible stiffness and the acceptable static displacement.

3.8.2 Passive Vibration Isolation The vibration source is the ground motion or the movement of the supporting structure. The purpose of passive vibration isolation is to reduce transmission of the vibration amplitude from the foundation or supporting structure to the superstructure or equipment. If the studied vibration amplitude is the displacement amplitude, what measures the vibration isolation effect is the displacement transmissibility. If the studied vibration amplitude is the acceleration magnitude, what measures the vibration isolation effect is the acceleration transmissibility. If the studied vibration amplitude is the velocity amplitude, the velocity transmissibility measures the vibration isolation effect. It is assumed that the mass of the isolated vibration m is born by the spring-damped system, which is placed on the base plate. Assume that the plate is subjected to simple harmonic vertical motion, the vibration displacement of the base (ground) is ug(t) = ug0 sin t, as shown in Figure 3.29. The absolute displacement of mass m is:

ut (t ) u(t ) ug (t )

(3.199)

Single Degree of Freedom Systems 117

m ug(t) k

c

Figure 3.29 Single degree of freedom passive vibration isolation system.

In the formula, u(t) is the relative displacement. The requirement for vibration isolation of the base is ut < ug, that is, the vibration of the equipment or structure is less than the vibration of the base. The relative displacement of mass u(t) can be obtained: 2

u(t )

Rdug 0 sin( t

)

(3.200)

n

The total displacement of a particle ut(t) is:

ut (t ) u(t ) ug (t ) ug 0 Rd 1

2 ( /

n

)

2

sin( t

1

) (3.201)

At this point, the displacement transmissibility Tn is:

Tu

u0t ug 0

Rd 1

2 ( /

1 1 ( /

2 ( / n

)2

2

n

n

)

)

2

(3.202)

2

2 ( /

n

)

2

It can be seen from (3.198) and (3.202) that the transmissibility of displacement is exactly the same as that of the force, indicating that the two vibration isolation problems are the same; therefore, their vibration isolation design methods are also basically the same. The curves plotted by the at different damping ratios are ratio of transmissibility to frequency n shown in Figure 3.30.

118

Structural Dynamics

The following phenomenon can be found from Figure 3.30: 1. Regardless of the damping ratio ζ, only when the frequency 2, can transmissibility be less than 1 and vibration ratio isolation effect will be achieved. The greater the frequency k / m, ratio is, the better the effect will be. Because of reducing the spring constant vibration isolator or appropriately increasing the mass should be chosen. When the frequency ratio > 5, the transmissibility decreases slowly, further reducing the stiffness or increasing the mass cannot significantly improve the effect of vibration isolation, so in 2.5~5 is often adopted. practice 2 , transmissibility is larger than 1. That is to say, 2. When the isolator amplifies the dynamic response instead, especially 1 (Resonance region) and the damping is relatively when small. The transmissibility reaches a great peak, and the isolator has the completely opposite effect. Attention must be paid to this and it must be avoided in vibration isolation design. When calculating the isolator parameters, the transmission rate according to the design requirements can first be determined. Then the frequency ratio and damping ratio ζ is determined. Finally, the stiffness k of the vibration isolation spring can be calculated. 30 25 20 Tu

ζ = 0.02 15 ζ = 0.05 10 ζ = 0.1 5 ζ = 1.0 0 0

0.5

ζ = 0.2 1

1.5

2

2.5 β

3

3.5

4

4.5

Figure 3.30 Relationship of transmissibility with different frequency ratio.

5

Single Degree of Freedom Systems 119 The isolation problem of building structures is similar to the single point system discussed above. For example, both of them attempt to improve the isolation efficiency by reducing the natural frequencies of the system. The difference is that the building structure is a multi-degree of freedom system, and the study of its isolation efficiency is more complex. Moreover, the ground motion is a wide-band vibration process, and there exists a frequency component with the same frequency as the natural frequency of the structure. Therefore, we can’t achieve the seismic isolation by avoiding the ground motion frequency. The structural isolation project covers a wide range, and it is one of the hot topics in the field of disaster prevention and mitigation engineering as well as structural engineering. Example 3.8 A reciprocating machine weighs 2 × 104N, It is known that when the machine operates at a speed of 40Hz, it can generate a vertical harmonic force of 500N amplitude. In order to limit the vibration of the machine to the building, the four corners of its rectangular bottom are supported by a spring. To reduce all the harmonic forces that the machine passes through the building to 80N, what spring stiffness is required? Solution: In this case, the force transmissibility is

TP

80 500

0.16 1

From formula (3.197), because of damping ratio ζ 0, TP then 2

2

W kg

7.25

Thus, the total spring stiffness is obtained: 2

k

W 7.25 g

4.51 105 N/m

2

1

6.25

120

Structural Dynamics

Therefore, the stiffness of each spring in the four supporting springs is:

ust

20 0.0444 m 451

It is also noted that the static displacement of the support springs due to the weight of the machine is:

ust

20 0.0444 m 451

Example 3.9 A machine is mounted on an elastic support, The measured natural frequency is fn = 12.5 HZ, damping ratio is ζ = 0.15. The mass that participate in vibrations is 880kg. It’s known that when the rotation speed of machine is N = 2400r / min, The magnitude of the unbalanced force is 1470 N. Determine the amplitude, force transmissibility, and the magnitude of force that passed to the foundation. Solution: From rotation speed N and natural frequency fn, we can obtain frequency ratio

n

2 N 60

1 2 fn

2400 60 12.5

3.2

The stiffness of the elastic support is

k m

2 n

880 (2

12.5)2

5.43 106 N

Thus, the amplitude of the vibration of the machine is

ust

P0 k (1

1 2 2

)

(2

)2

1470 1 5.43 106 (1 3.22 )2 (22 0.15 3.2)2

0.0000291m 0.0291mm

Single Degree of Freedom Systems 121 Force transfer rate is

Tp

1 (2 0.15 3.2)2 (1 3.22 )2 (2 0.15 3.2)2

0.149

Thus, the amplitude of the force transmitted to the foundation is

PT

P0 Tp 1470 0.149 219N

3.9 Structural Vibration Induced Fatigue With the rapid development of modern industry, many machines and equipments are developing towards high speed, high temperature and high pressure. The factors that cause the failure of engineering structures are also increasing, in which vibration and fatigue are two important factors. Vibration hazards are manifested in the following aspects: failure of instruments and equipment, ergonomics and health loss of personnel, vibration intensity of structures, vibration reliability of complex equipment systems, noise and pollution caused by vibration, etc. It is easy to be overlooked in many circumstances, where there are many accidents. In the past half-century, many scholars have carried out a series of research studies on the structural vibration induced fatigue problem. Many achievements have been made, but a large number of results are still distributed in scattered literature, which makes it difficult to form a system for the structural vibration induced fatigue research. Therefore, understanding the definition and characteristics of structural vibration induced fatigue is helpful for further study in structural vibration induced fatigue.

3.9.1 Definition of Vibration Induced Fatigue With the increasing demand for the accuracy of structural fatigue design, fatigue research has gradually developed from material mechanics to elasticplastic mechanics and fracture mechanics. Up to now, there is no consensus on the understanding of vibration induced fatigue in academia and engineering circles, but great importance has been attached to this problem in practice. Vibration induced fatigue involves the dynamic characteristics of structures, so it is very

122

Structural Dynamics

difficult to define it accurately. Although many scholars have described the definition of vibration induced fatigue, these definitions do not reveal the dynamic nature of vibration induced fatigue. Although the term “vibration induced fatigue” often appears in books and other existing literature, even in some standards and specifications, it is not well explained.

3.9.2 Characteristics of Vibration Induced Fatigue It is important to know the type of the problem before studying the specific problem. According to the different types of excitation, vibration induced fatigue can be divided into tension-compression vibration induced fatigue, torsional vibration induced fatigue and bending vibration induced fatigue. On the contrary, it is called non-resonant fatigue. According to the ratio of excitation frequency to structure fundamental frequency, vibration induced fatigue is divided into high frequency vibration induced fatigue and low frequency vibration induced fatigue. 1. Resonant fatigue and non-resonant fatigue Resonance is a synthetical equilibrium phenomenon between external force and structural inertia force, elastic force and damping force. It is characterized by modal inertia force and damping force occurring in the structure. Damping force is an important factor to determine the size of resonance response, so resonance fatigue depends on the distribution characteristics of strain modes which play a major role. Resonance types include global resonance, component resonance and local resonance. Resonance fatigue is more related to component resonance or local resonance. Some vibration excitations often cause local mode and excitation vibration coupling, and the damage site is usually in the local resonance where strain is large and defective or stress concentration. Fatigue extensively exists in large-scale structures excited by impact, transient or random vibration. Non-resonant fatigue is that the excitation frequency of the structure is far from the resonant frequency of the structure. It often exists in the case of single-frequency vibration excitation or structural stiffness is large but the excitation frequency is low. The amplitude of structural dynamic response in resonance is much larger than that in non-resonance under the same magnitude of excitation. The amplitude of resonance fatigue response mainly depends on the magnitude of excitation and damping. A large number of medium-

Single Degree of Freedom Systems 123 magnitude excitations induce resonance fatigue failure. Controls of stiffness and damping, mainly contributing to fatigue failure, are small and larger orders of excitation. 2. High frequency vibration induced fatigue and low frequency vibration induced fatigue The low frequency vibration induced fatigue mainly studies the fatigue problem that the excitation frequency is less than half of the fundamental frequency of the structure. Its characteristic is that the damping and inertia of the system are very small, the elasticity of the system plays a major role, and the system is similar to a pure elastic structure. Therefore, the low frequency vibration induced fatigue failure is mainly controlled by the elastic strain. The characteristics of fatigue are that the damping and elasticity of the system are very small, and the inertia of the system plays a major role. The system is similar to a pure inertia structure for quality control. In engineering, some large-scale structures (such as aircraft) may work in a wide-band excitation environment. Aircraft structures may always remain resonant with the fatigue growth, which results in frequencyband excitation fatigue. Considering the influence of vibration nonlinearity, the vibration response of the structure may also have some nonlinear characteristics, such as self-excited vibration. Dynamic, parametric vibration, multi-frequency response, jumping and synchronization phenomena are prone to fatigue problems that cannot be explained by linear vibration theory.

Exercises 3.1 In the undamped and damped free vibration, what factors are associated with the natural frequencies and dynamic responses? 3.2 As the dynamic system shown in the picture, using jack to make m produce lateral displacement of 30 mm, then the jack is suddenly released, the system begins to vibrate freely. The lateral vibration was measured 10 mm after 8 periods. Determine: (1) Damping ratio and damping coefficient of the structure (2) Amplitudes after 20 weeks of vibration (3) If the lateral displacement is 15mm after 8 periods of vibration, while the damping and mass remain constant, the stiffness must be set at what value?

124

Structural Dynamics u0 m

3m

k

k

c

3.3 Single degree of freedom system, known m = 100kg, k = 6000 N / m. Free vibration begins when initial condition u(0) 20mm, u(0) 0.1 m/s. Determine: (1) The natural frequency and period of the system (2) The displacement and velocity when t = 1.0s for two cases of c = 0, c = 200N.s / m. 3.4 The basic power system shown here has the following characteristics: m = 20kg, k = 3 ×104 N / m. If the system is subjected to a resonance harmonic load( = n) and the initial state is static, specify the value of the dynamic magnification factor Rd(t) after five cycles( t = 10 ), (1) Undamped system (2) ζ = 0.02; (3) ζ = 0.05. O

u(t)

k m

P(t)

c

3.5 What are the methods for calculating damping ratio? 3.6 What are the types of impact loads, and what are the basic ideas and procedures for their dynamic responses? 3.7 Use MATLAB language to draw up a procedure that solves structural dynamic response (displacement response and acceleration response) with Duhamel integral. Single degree of freedom, mass m = 2 ×104 kg, stiffness k = 3 × 106 N / m, is subjected to 0.2g, El, Centro seismic waves. 3.8 What are the basic steps in the frequency domain analysis of structural dynamic responses? 3.9 What are the structural dynamic tests? What are the common excitation and sensing devices?

Single Degree of Freedom Systems 125 3.10 A reciprocating machine weighs 2 × 104N; it is known that when the machine operates at a speed of 40Hz, a vertical harmonic force of 1000N amplitude is generated. In order to limit the vibration of the machine to the building, the four corners of its rectangular bottom are supported by a spring. What is the spring stiffness required to limit all the harmonic forces that pass from the machine to the building to the 100N?

References 1. Clough, R.W. and Penzien, J., Dynamics of structures//Dynamics of Structures, 2005. 2. Chopra Anil, K., Dynamics of Structures, Prentice Hall, 2000. 3. Wilson, J.F., Dynamics of Offshore Structures, 1984. 4. De Wachter, S., Tzavalis, E., Detection of structural breaks in linear dynamic panel data models. Comput. Stat. Data Anal., 56, 11, 2012. 5. Cluni, F., Gioffrè, M., Gusella, V., Dynamic response of tall buildings to wind loads by reduced order equivalent shear-beam models. J. Wind Eng. Ind. Aerod., 123, 2013. 6. Mackie, R.I., Dynamic analysis of structures on multicore computers – Achieving efficiency through object oriented design. Adv. Eng. Softw., 66, 2013. 7. Birman, V., Elisakoff, I., Singer, J., On the effect of axial compression on the bounds of simple harmonic motion, 1982. 8. Zavodney, L.D. and Nayfeh, A.H., The response of a single-degree-offreedom system with quadratic and cubic non-linearities to a fundamental parametric resonance. J. Sound Vib., 120, 1, 63–93, 1988. 9. Zhao-Dong, X., Le-Wei, M., Dynamics of Structures, Science Press, Beijing, 2007.

4 Multi-Degree of Freedom System The single degree of freedom system has previously been described in detail. Although most of the structures in engineering practice can’t be reduced to a single degree of freedom system to calculate, simplifying the structures as a single degree of freedom system will bring about greater error. We can simplify the water tower and single-story building into a single degree of freedom system, but it cannot be done for multi-story buildings. It should be noted that a single degree of freedom system and a single mass point system are two different concepts. The reason we say water towers and single-story buildings are not only single-point system but also a single degree of freedom system is that the mass of the two buildings is mainly concentrated at the top, so they can be simplified into a single-point system by using the centralized mass method. On the other hand, for a structure, if we only consider the certain horizontal direction (such as x-direction) vibration and disregard the vertical deformation or y-direction vibration, it can also be called a single degree of freedom system. It should be noted that not all of the single-point systems are single degree of freedom systems. As shown in Figure 4.1, the single point m contains two degrees of freedom of horizontal and vertical, and therefore should fall under the multiple degree of freedom system. In short, for the mass decentralized structure system, in order to reflect its dynamic performance more accurately, the structure system can be simplified as a multi-particle system. For example, for a multi-story frame with rigid floors, the mass can be concentrated at the floors of each story. For multispan, unequal height, single-story plants, the mass can be concentrated at the roofs of each span. For other structures, such as chimneys, it can also be divided into several sections according to the calculation requirement, and then every section can be converted into a particle to analyze (see Figure 4.2). When a multi-point system is sufferedone-way vibration merely, the number of mass points equals to that of degree of freedom. It should be pointed out that for the frame structure, the main purpose of emphasizing the rigid floor or shear frame assumptions is to ensure that the simplified particle only has horizontal displacement without rotation degrees of freedom. Yong Bai and Zhao-dong Xu (eds.) Structural Dynamics, (127–205) © 2019 Scrivener Publishing LLC

127

128

Structural Dynamics

m

Figure 4.1 Multiple degree of freedom system with one point.

(a) Multi-layer frame

(b) Unequal height plant

(c) Chimney

Figure 4.2 Multi-degree of freedom system.

4.1 Two Degrees of Freedom System Although the two degrees of freedom system is the simplest special case of the multi-degree of freedom system, the analytical method applied in the two degrees of freedom system is universal in the multi-degree of freedom system. Therefore, the two degrees of freedom system is introduced firstly.

4.1.1 Establishment of Motion Equation of Undamped Free Vibrations Similar to the single degree of freedom system, the dynamic balance method and the virtual displacement principle described in Chapter 2 are also applicable to the multi-degree of freedom system. Here, only the dynamic balance method is used as an example to illustrate its application in multi-degree of freedom systems. For that, an inertial force can be assumed to be exerted on each particle (n all), which transforms the multi-degree of freedom system into n single degree of freedom systems, and treats the dynamic problem as a static equilibrium problem to solve. The concrete application can be divided into the flexibility method and

Multi-Degree of Freedom System

129

stiffness method. Here the flexibility method will first be briefly described, then the way to establish motion equation by the stiffness method will be introduced in detail. For the two degrees of freedom system shown in Figure 4.3(a) and (b), the displacement u1(t) and u2(t) of the masses m1 and m2 can be regarded as the static displacement generated by combined action of the inertial force FI1 = –m1ü1(t) and FI2 = –m2ü2(t). Apply overlay principle results in:

u1 (t ) FI1Δ11 FI2Δ 21

m1u1 (t )Δ11 m2u2 (t )Δ12

u2 (t ) FI1Δ12 FI2Δ 22

m1u1 (t )Δ 21 m2u2 (t )Δ 22

(4.1)

In which Δ11, Δ12, Δ21, and Δ22 are the flexibility coefficients of the structure, the physical meaning of Δij is the displacement of point i generated by the unit force acted at point j (shown in Figure 4.3(c), (d)), according to virtual principle, Δ12=Δ21. The above formula contains the flexibility coefficient, so the above-mentioned way to establish the displacement equation is also called the flexibility method. It can be seen that the flexibility method is used to establish the motion equations by forming the displacement equations. The way to establish the motion equation of the two degrees of freedom system by the dynamic balance method of stiffness method will also be introduced. Figure 4.4 is a shear-type frame structure simplified as a twodegree-free system, when subjected to transient load, it begins to perform free vibration, taken the particle 1 as a separator part, the effect of the inertial force F11 and elastic recovery force FS1 subjected on it are respectively:

FI1

m1u1

FS1 k2 (u2 u1 ) k1u1

Figure 4.3 Establishment of the equation of motion by elasticity method.

(4.2) (4.3)

130

Structural Dynamics u2

FI2

Fs2

m2

k2(u2–u1)

k2 u1

FI1

m1

Fs1 m1

k1

k1u1

Figure 4.4 Dynamic balance of two degrees of freedom system.

Similarly, for point 2:

FI 2 FS2

m2u2

(4.4)

k2 (u2 u1 )

(4.5)

It should be noted that the negative signs at the right end of FI1, FI2, and FS2, which indicate that the force is opposite to the corresponding displacement direction. According to the dynamic balance equation FI+FS=0, the motion equations about m1 and m2 are:

m1u1 k2 (u2 u1 ) k1u1

0

m2u2 k2 (u2 u1 ) 0

(4.6)

After arranged, there are:

m1u1 (k1 k2 )u1 k2u2 m2u2 k2 (u2 u1 ) 0

0 (4.7)

Multi-Degree of Freedom System

131

1 k22

k21 k2

k2

k2

k2 k11 0

k1

k1

k12 0

1

Figure 4.5 The physical meaning of the stiffness factor.

Written in the matrix form, there are:

M

In which, [ M ]

u

m1 0 0 m2

K

u

, [K ]

0

(4.8)

k11 k12

k1 k2

k2

k21 k22

k2

k2

, the

physical meaning of kij is the counter-force of point j generated by the unit displacement of point i, as shown in Figure 4.5. Since kij represents the lateral stiffness of the system, the method of establishing the motion equation described above is also called the stiffness method.

4.1.2 Natural Frequency and Vibration Mode Shape The solutions of the differential equations group (4.8) are:

u1

X1 sin(

n

t

)

u2

X 2 sin(

n

t

)

(4.9)

In which, ωn is the self-oscillation frequency, ϕ is the initial phase angle, and X1 and X2 are the displacement amplitudes of the particles 1 and 2, respectively.

132

Structural Dynamics

Substituting equation (4.9) into equation (4.7): 2 n

(k11 m1

)X1 k12 X 2

k21 X1 (k22 m2

2 n

0

)X 2

(4.10)

0

The above equation is a homogeneous equations group about X1 and X2. Although X1=X2=0 is the solution of the equations group, it only means that the system is in a static state, therefore it is not the solution of free vibration. According to the Cramer rule in linear algebra, if the homogeneous system has a nonzero solution, its coefficient determinant must be zero.

k11 m1

2 n

k12

k21

k22 m2

0

2 n

(4.11)

Expanding it:

(

2 2 n

)

k11 m1

k22 m2

2 n

k11k22 k12 k21 m1m2

0

(4.12)

Solving it:

2 n

1 k11 2 m1

k22 m2

1 k11 2 m1

k22 m2

2

k11k22 k12 k21 (4.13) m1m2

These two roots are positive roots; this is because if the two roots are negative real or complex number, then the ωn1, ωn2 should be imaginary or complex number, and the u1 and u2 obtained by substituting them into the equation (4.9) will be a hyperbolic function containing the time parameter, which is obviously contradictory to that the system is stable in the equilibrium position (t = 0) under the effect of restoring force. Now the two positive roots of the ωn can be obtained, knowing the frequency ωn of the system through Eq. (4.13) is related only to the parameters (mass and stiffness) of the system itself, but not to other conditions. The smaller one is called the first self-resonant circular frequency or basic natural frequency

Multi-Degree of Freedom System

133

(fundamental frequency), the larger one is called the second self-resonant circular frequency. What should be noted is that ωn in this book indicates the natural frequency of the system, ωni indicates the i-th order natural frequency of the system, ω indicates the frequency of the system’s excitation. According to the Cramer rule, the solution of Equation (4.10) is not independent because the coefficient determinant of Equation (4.10) is equal to zero. Assuming that the corresponding solutions of the ωni are Xi1 and Xi2 (the first subscript indicates the i-th order frequency and the second subscript represents the corresponding particle). Although the solution of the system (the displacement amplitude Xi1, Xi2) is not independent, the ratio is constant. Corresponding to ωni

Xi 2 X i1

m1

2 ni

k11

k12

k1 k2 m1 k2

2 ni

i 1, 2

,

(4.14)

The particle displacements obtained by equation (4.9) are: Corresponding to ωn1

u11

X11 sin(

n1

t

1

)

u12

X12 sin(

n1

t

1

)

u21

X 21 sin(

n2

t

2

)

u22

X 22 sin(

n2

t

2

)

(4.15)

Corresponding to ωn1

(4.16)

Substituting equation (4.15) and (4.16) into equation (4.14), respectively

ui 2 ui1

Xi 2 X i1

m1

2 ni

k12

k11

k1 k2 m1 k2

2 ni

i 1, 2

(4.17)

It can be seen that the displacement ratio of the two particles in the vibration process is not only time-independent but also constant. In other words, at any time during the structural vibration process, the displacement ratio of the two particles is always constant. This type of vibration is

134

Structural Dynamics

usually called the main vibration mode shape, vibration mode for short, which is only related to the parameters of the system itself and is irrelevant to initial condition. The vibration mode shape corresponding to ωni is called the i-th vibration mode shape. In addition, since the main mode depends only on the relative value between the mass displacements, generally, the displacement value of one of the particles (usually the first or last) is usually set to 1 for the sake of simplicity. This process is called normalization of vibration mode shape. In addition, for the first mode, it can be proved that the displacement ratio is always greater than zero, which means that the vibrations of m1 and m2 possess same phase and same direction. For the second mode, the ratio is always less than zero, which means that the vibration of m1 and m2 possess reverse phase and opposite direction, so there is a point on the curve that vibration does not occur; this point is called the node.

4.1.3 General Solutions of the Equations of Motion According to the theory of homogeneous linear equations solution space, the general solution of the equations group (4.7) should be a linear combination of the fundamental solution systems (4.15) and (4.16):

u1 (t )

1

X11 sin(

n1

t

1

)

2

X 21 sin(

n2

u2 (t )

1

X12 sin(

n1

t

2

)

t

1

)

2

X 22 sin(

n2

t

2

)

(4.18)

In which, α1 and α2 are the combination coefficient, which can be contained in the displacement amplitude X, so the above formula can be simplified as

u1 (t ) X11 sin(

n1

t

1

) X 21 sin(

n2

u2 (t ) X12 siin(

n1

t

2

)

t

1

) X 22 sin(

n2

t

2

)

(4.19)

From Equation (4.17), it is clear that the ratio of displacement amplitude X is constant, so the above equation contains only four arbitrary constants, X11, X21 (this is because X21, X22 can be obtained by X11, X21 according to the ratio of displacement amplitude), ϕ1, and ϕ2. They can be determined by the initial condition u1 (0), u2 (0), u1 (0) and u2 (0) of the vibration. It can be seen that the vibrations of each particle represented by Eq. (4-19) are the synthesis of two simple harmonic vibrations. It is not a simple harmonic vibration certainly. Only when the ratio of the initial displacement of each

Multi-Degree of Freedom System

135

particle is equal to that of the main mode and the initial speed is equal to zero, can the system follow the main vibration mode. Under normal initial conditions, the system does not implement simple harmonic vibrations, and the ratio of displacement between particles is no longer constant and its value will change over time.

4.2 Free Vibrations of Undamped System 4.2.1 Establishment of Motion Equation We can generalize the form of the two degrees of freedom equations of motion to a multi-degree of freedom system. In order to facilitate expression and computation, being similar to equation (4.8), the motion equations of the undamped multi-degree of freedom system are expressed in the form of the following matrix:

[M] u

[K ] u

0

(4.20)

In which,

m1 m2

[M]

k11 k12 ... k1n

0

k21 k22 ... k2n

,[K ]

...

... ... ... ... kn1 kn 2 ... knn

mn

0

u1 u2

, u

... un

Now a five degrees of freedom system (shown in Figure 4.6) is served as an example, the physical meaning of the coefficient kij contained in the stiffness matrix [K] is shown as follows:

K

5 5

k1 k2

k2

0

0

0

k2

k2 k3

k3

0

0

0

k3

k3 k4

k4

0

0

0

k4

k4 k5

k5

0

0

0

k5

k5

(4.21)

136

Structural Dynamics

k5 k4 k3 k2

k1

Figure 4.6 Five degrees of freedom of system.

Being similar to equation (4.9), the solution of the equations group (4.20), the displacement of each particle can be expressed as follows:

u(t )

X sin(

n

t

)

(4.22)

In which:

u1 (t ) u(t )

X1

u2 (t )

X2

, {X }

Xn

un (t )

Substituting (4.22) into equation (4.20) and noting that it is arbitrary results in: 2 n

(K

M ) X

0

(4.23)

The condition that the above equation possesses the nonzero solution is:

K

2 n

M

0

(4.24)

Multi-Degree of Freedom System

137

The natural frequency of the system can generally be obtained by solving the generalized eigenvalues of equation (4.23). According to the linear algebra, we can see that when the mass matrix [M] is positive definite (the order principal minor determinant of [M] are all greater than zero), the stiffness matrix [K] is also positive or semi-definite (the all order principal minor determinant of [K] are greater than or equal to zero), then all the roots of the equation (4.23) of n2 are positive real or zero. Those roots ( n2 ) are called the eigenvalue of the above equations group. ωn is also called the natural frequency of the system. Generally, the natural frequencies are arranged from small to large, the smallest of them is called the first frequency or fundamental frequency of the system, and the corresponding period T1 = 2π / ωn1 is called the first cycle or the basic period, since the first frequency is the minimum frequency of the system, it means that the basic period should be the longest period. There are n natural frequencies for n degrees of freedom. Those natural frequencies are only related to the parameters of the system itself, independent of other parameters such as external loads, boundary conditions, and so on.

4.2.2 Vibration Shape and Its Orthogonality Since the coefficient determinant (4.24) of the equations group (4.23) is equal to zero, the solution {X} of the equations group (4.23) is not independent. Therefore, it is necessary to substitute the obtained natural frequency into equation (4.23) and let Xij=1 (generally take j = 1 or j = n), so that the equations (4.23) can be transformed into a non-homogeneous algebra group, the solution of the above equations, we can get the i-type main vibration shape corresponding i-order frequency ωni by solving the above equations.

X i1 X

Xi 2 i

X i1 X i 2

Xin

T

(4.25)

Xin According to the linear algebra, we can see that the above-mentioned i-th main mode is the eigenvector corresponding to one of the eigenvalues 2 ni of the equations group (4.23).

138

Structural Dynamics

The so-called orthogonality of vibration shape refers to the following relationship existing between any two different mode vectors {X} i and {X}j corresponding to the two different frequencies ωni and ωnj in a multi-degree of freedom system or an infinite degree of freedom system.

X

X

T i

T i

[M] X

[K ] X

j

j

0, i

j

(4.26)

0, i

j

(4.27)

0

(4.28)

Now it is proved as follows. Proof Method 1: According to equation (4.23):

(K

K

X

2 ni

M ) Xi

i

2 ni

M X

(4.29)

i

And the transpose vectors of the main vibration shape of the j-th order are simultaneously applied to both sides:

X

T

K X

j

i

2 ni

X

T j

M X

i

(4.30)

Noting that:

[K ]T

[K ], [ M ]T

[M]

(4.31)

According to the transitive rule of the linear algebra, the two ends of the equation (4.30) are transposed at the same time:

X

T j

[K ] X

j

2 1

X

T i

[M] X

j

(4.32)

Multi-Degree of Freedom System

139

Similarly, corresponding to the frequency ωnj, both ends of the formula T (4.28) are previously multiplied by {X}j first and multiplied by X i second, which can obtain a resembling equation (4.30):

X

T i

K X

2 nj

j

X

T

M X

i

(4.33)

j

Subtracting equation (4.33) from equation (4.32) results in:

( Due to ωni

2 ni

2 nj

){ X }Ti [ M ]{ X } j

0

(4.34)

ωnj:

{ X }Ti [ M ]{ X } j

0

(4.35)

Similarly, the orthogonality of the stiffness matrix [K] can be proved. Certainly, there are many ways to prove the above theorem, such as the equivalence law of work of Betti’s law, as shown in Proof Method 2. Proof Method 2: According to the equivalence law of work, the work done by one set of loads on the deflections due to a second set of loads is equal to the work of the second set of loads acting on the deflections due to the first. Now the above-mentioned law is illustrated by the orthogonality between the first and second vibration shape of the three degrees of freedom, in which the mode is used as the displacement and the inertia force, m n2 X , as the force. The situation corresponding to the first mode is assumed as the first state (Figure 4.7a); the situation corresponding to second mode is assumed as the second state (Figure 4.7b). Then the work done on the second state displacement by the first state force is:

W12

2 n1

m1 X11 X 21

2 n1

2 n1

m2 X12 X 22

m3 X13 X 23

m1 2 n1

X11 X12 X13

X 21 m2

X 22 m3

2 n1

X

T 1

M X

2

X 23 (4.36)

140

Structural Dynamics X13

X23

m3 ωn12X13

m3 ωn22X23

X22

X12

m2 ωn22X22

m2 ωn12X12

X21

X11 m1 ωn22X21

m1 ωn12X11

(a)

(b)

Figure 4.7 The reciprocity theorem of work.

And the work done on the first state displacement by the second state force is:

W21

2 n2

m1 X 21 X11

2 n2

2 n2

m2 X 22 X12

m3 X 23 X13

m1 2 n1

X 21 X 22 X 23

X11 m2

X12 m3

2 n2

X

T

M X

2

X13 (4.37)

1

According to Betti’s law of work reciprocity, W12 = W21, results in: 2 n1

X

T 1

M X

2 n2

2

X

T 2

M X

1

Since equation (4.37) can also be rewritten as follows:

W21

2 n2

2 n2

2 n2

2 n2

2 n2

2 n2

m1 X 21 X11

m1 X11 X 21

2 n2

X

T 1

[M] X

m2 X 22 X12 m2 X12 X 22

2

m3 X 23 X13

m3 X13 X 23

(4.38)

Multi-Degree of Freedom System

141

So: 2 n2

T

X

[M] X 2

2 n2

1

T

X

[M] X

1

(4.39)

2

So equation (4.38) can be written as:

( 2 n2

Because of

2 n2

2 n1

2 n1

) X

T 1

[M] X

2

0

(4.40)

: T

X

1

[M] X

0

2

Prove finished Proof Method 3: When the multi-free motion equation is established by the flexibility method:

(

2 n

[ ][ M ] [1]) X

0

(4.41)

In which, the flexibility coefficient matrix of system [ ] = [K]–1. Expanding the above formula in the form of equations group, corresponding to the frequency ωnj and amplitude of Xji:

m1

11

2 nj

1 X j1 m2

m1

21

2 nj

X j1

m1

n1

2 nj

X j1 m2

m2

22

n2

2 nj

X j2

mn

1n

2 nj

X jn

0

1 X j2

mn

2n

2 nj

X jn

0

1 X jn

0

12 2 nj

2 nj

X j2

mn

nn

2 nj

(4.42)

After deforming the above equation, the expression of the amplitude X corresponding to ωnj and ωnk can be expressed as:

142

Structural Dynamics

X kn

2 nk

2n

X kn

2 nk

mn

nn

X kn

2 nk

X j2

mn

1n

X jn

2 nj

22

X j2

mn

2n

X jn

2 nj

n2

X j2

mn

nn

X jn

2 nj

X k1

m1

11

X k1 m2

12

Xk2

mn

1n

Xk2

m1

21

X k1 m2

22

Xk2

mn

X kn

m1

n1

X k1 m2

n2

Xk2

X j1

m1

11

X j1 m2

12

X j2

m1

21

X j1 m2

X jn

m1

n1

X j1 m2

(4.43)

(4.44)

Multiplying the first formula, second formula ... n-th formula of the equation (4.43) by m1 nj2 X j1 , m2 nj2 X j 2 , mn nj2 X jn respectively, and then adding them together, similarly, multiplying the first formula, second formula and 2 2 2 n-th formula of the equation (4.44) by m1 nk X k1 , m2 nk X k 2 , mn nk X kn respectively, and then also adding them together, obviously, the right of the two above equations is exactly identical, so the left should also be equal, resulting in: 2 nj

m1 m1

X j1 X k1 m2 2 nk

Because of ωnj

X j1 X k1 m2

2 nj

X j 2 Xk2 2 nk

mn

X j2 Xk2

2 nj

mn

X jn X kn 2 nk

X jn X kn

(4.45)

ωnk, it can be obtained that:

m1 X j1 X k1 m2 X j 2 X k 2

mn X jn X kn

0

(4.46)

The above equation is equation (4.35), proof finished.

4.2.3 Generalized Mass and Generalized Stiffness For the equations (4.26) and (4.27), the precondition of the mode orthogonality is i j, if i = j, according to the property of determinant operation, T T X i [ M ] X j and X i [K ] X j will be unequal to zero. Mk and Kk are

Multi-Degree of Freedom System

143

called generalized mass and generalized stiffness respectively corresponding to the k-th mode, defined as follow:

X X

T k

[M] X

T k

[K ] X

k

Mk

(4.47)

k

Kk

(4.48) T

Both sides of the equation (4.29) are multiplies by X i , it becomes:

X

T i

[K ] X

2 ni

i

X

T i

[M ] X

i

(4.49)

According to the definition of generalized mass and generalized stiff2 ness, the above equation is K i ni M i , therefore:

Ki Mi

ni

(4.50)

It can be seen that the corresponding frequency can be obtained directly by generalized mass and generalized stiffness. For the n degrees of freedom system, the n×n order square matrix can be formed by ranging the obtained n modes {X}i from left to right, this square matrix is called the vibration matrix. That is:

[X ]

X 1, X

2

,

, X i,

, X

X11

X12

X1 j

X1n

X 211

X 22

X2 j

X 2n

X j1

X j2

X jj

X jn

Xn1

Xn 2

X jn

Xnn

With this substituted into Eq. (447), it becomes:

n

(4.51)

144

Structural Dynamics X X X

T

T

m1

1 T

m2

2

[M] X X X [ X

1

T

mj

j

mn

T n

, X

2

,

, X

i

,

, X

n

(4.52)

]

M1 M2 Mj Mn

The above equation is called the generalized mass matrix. Similarly, the generalized stiffness matrix can be obtained by using the modal matrix. That is: X X X

T

T 1

k11

k12

k1 j

k1n

k21

k22

k jj

k jj

k j1

kj2

k jj

k jn

kn1

kn 2

knj

knn

T 2

[K ] X X X [

X

T j T n

1

, X

2

,

, X

i

,

, X

n

]

(4.53)

K1 K2 Kj Kn

Example 4.1 As shown in Figure 4.8, a two-layer shear-type frame structure, each layer of mass is known, which is m1 = 60t, m2 = 50t, the first layer’s lateral stiffness is k1 = 5×104kN/m, the second floor of the lateral stiffness is

Multi-Degree of Freedom System

145

k2 = 3×104kN/m. Solve the problem of the system’s natural frequency and vibration of the structure, and verify the orthogonality of its main mode. X12=1.0

X22=–1.0

m2, k2 X11=0.488

X11=1.710

m1, k1

(a) Frame

(b) The first vibration shape

(c) The second vibration shape

Figure 4.8 Two-layer shear frame structure and vibration mode.

Solution: According to (4.8), the inter-layer stiffness coefficients of the frame layers can be obtained respectively:

k11

k1 k2

k12

k21

k22

k2

5 104 3 104

8 104 kN/m

3 104 kN/m

k2

3 104 kN/m

From equation (4-12), the frequency equation is given as follows

8 104 60 3 104

2

3 104 3 104 50

2

0

Expanding the above equation, there are 0.00003 n4 0.058 n2 15 0 Solving the above equation, there are ωn1 = 17.54rad/s, ωn2 = 40.32rad/s. Now, the corresponding periods are respectively:

T1

2 /

n1

2 / 17.54 0.358s, T2

According to equation (4-17): X For the first vibration shape: 12 X11 1 0.488

m1

2 n1

k12

k11

0.156s 60 17.54 2 8 104 3 104

146

Structural Dynamics

For the second vibration shape:

X 22 X 21

2 n2

m1

60 40.322 8 104 3 104

k11

k12

1 1.710

The above modes are shown in Figures 4.8(b) and (c), respectively. Next, the orthogonality of the main mode will be checked, for the mass matrix and stiffness matrix, according to equation (4.26) and equation (4.27), it can be obtained that:

X

T i

[M] X

k

T

X

1

[M ] X

60 0 X

T i

[K ] X

k

X

T 1

2

0 50

[K ] X

8 3

3 3

{0.488 1} 1..710 1

2

0

{0.488 1} 1..710 1

0

The generalized mass and generalized stiffness are given by (4.47) and (4.48) as follows:

X X X X

T i

[M] X

T i T 2 T 2

[K ] X

[M ] X [M ] X

64.29t

1

M1

1

K1 1.977kN/m

2

M2

225.45t

2

K2

36.65 kN/m m

4.3 Practical Calculation Method of Dynamic Characteristics The dynamic characteristics of the structure consist of the physical quantities such as the natural frequency, period, and mode of the structure. In the dynamic analysis of the structure, determining the dynamic characteristics

Multi-Degree of Freedom System

147

of the system should be done first. If the dynamic characteristics are calculated only by analytic method, the calculated workload will be very heavy when the degree of freedom of the system exceeds three. For example, from the formula (4.13), even for a two degrees of freedom system, the analytical formula of its natural frequency has been very complicated. In addition, it is even unable to be solved using the exact method for some complex structure sometimes. So, in order to effectively deal with such a wide range of computing problems, we need to resort to the simplicity and accuracy of the approximate calculation method, which is the reason why dynamic characteristics of the practical calculation method is proposed. Here, some conventionally used calculation methods will be introduced.

4.3.1 Dunkerley Formula It is important to estimate the fundamental frequency of the system quickly in the dynamic analysis. Therefore, the Dunkerley formula for estimating the first frequency is introduced. Taking the three degrees of freedom system as an example, the system characteristic equation established by the softness method is:

1

m1

11

m2

m3

12

2 n

m1

22

m2

m1

32

21

13

1

m3

m2

31

0

23

2 n

(4.54)

1

m3

33

2 n

In which, ij is the flexibility coefficient of system, the expanded formulary of which is a cubic algebraic equation about 1 . 2 n

1

3

(

2 n

m1

11

m2

m3 )

22

33

2

1

0

2 n

Assuming that the three roots of the equation are equation can be written as:

1 2 n

1 2 n1

1 2 n

1 2 n2

1 2 n

1 2 n3

1 2 n1

0

,

(4.55) 1 2 n2

1 2 n3

, an

(4.56)

148

Structural Dynamics

Expanding it: 3

1

1

2 n

1

2 n1

1

2 n2

2

1

2 n3

0

2 n

(4.57)

Comparing equation (4.57) with equation (4.55) results in:

1

1

2 n1

1

2 n2

2 n3

m1

m2

11

22

m3

33

(4.58)

Since the high frequency part of the engineering practice is much higher than that of the fundamental frequency, the approximate formula for the first frequency of the system can be obtained by ignoring the high order 1 at the left end of the above equation. frequency 1 2 n2

2 n3

1 2 n1

m1

m2

11

22

m3

33

(4.59)

It is the basic frequency calculation formula given by Dunkerley, which is the general form of Dunkerley formula for n degrees of freedom system.

1 2 n1

Because of

ii

mi 1 /

2 nii

n ii

mi

(4.60)

i 1

, the above equation can also be written as:

1 2 n1

n

ii 1

1 2 nii

(4.61)

Dynamic analysis often requires a quick estimate of the fundamental frequency of the system when the quality or stiffness parameters of the system changes. The Dunkerley formula can give a convenient calculation for this. If the fundamental frequency of the original multi-degree of freedom system is ωn1 and the increment of each mass is Δmi, then according

Multi-Degree of Freedom System to the Dunkerley formula, the base frequency mass increased is

1 2 n1

n

n ii

of the system after the

n

(mi Δmi )

i 1

n1

ii

mi

i 1

ii

Δmi

i 1

149

n

1 2 n1

ii

Δmi

i 1

(4.62) According to the formula (4.59), it is obvious that the Dunkerley formula is accepted by subtracting the high frequency term at the left end, so the obtained fundamental frequency is lower than the actual value. In addition, the result from the Dunkerley formula is more precise when the frequency of the higher-order mode of the strucyure differs greatly from its fundamental frequency.. Contrarily, the deviation of Dunkerley formula is greater when frequencies of structure are close. So, for the dense spectrum of large-span bridges, large-span grids, and other structures, the calculation obtained by the Dunkerley equation is less accurate. Example 4.2 For the simply supported beam shown in Figure 4.9, if ml m1 m2 m3 , find the fundamental frequency of the system by the 3 Dunkerley formula. m1

l/4

m2

EI

l/4

m3

l/4

l/4

l

Figure 4.9 Three-particle simply support beam structure.

Solution: According to the figure multiplication, it can be obtained that 25l 3 81l 3 the flexibility coefficient of system 11 33 , 22 , sub3888 EI 3888 EI stituting them into the formula (4-59):

1 2 n1

25l 3 2 3888 EI

ml 3

81l 3 3888EI

ml 3

150

Structural Dynamics

9.44 EI , the more accurate result is Solving it, results in n1 m l2 9.86 EI with the deviation is -4.15%. n1 m l2 Example 4.3 As shown in Figure 4.10, it is known that the masses of the unequal high single-story plant are m1 = 59200N, m2 = 50000N. The soft= 2.13×10–4m/kN, = 5.65×10–4m/kN. In addiness coefficients are tion, the base frequency of the plant has been obtained, ωn1 = 5.16(rad/s). If the mass m1 of the plant has increased by 20000N, estimate the base frequency of the plant after the mass changed. m m

Figure 4.10 Unequal height single story plant.

Solution: According to equation (4-62),

1 2 n1

1 2 n1

n ii

Δmi

i 1

1 5.162

20 2.13 10

4

Solving it, n1 4.90 rad/s . The exact solution after the parameter changed is 4.95s-1, and the estimated error is -1%.

4.3.2 Rayleigh Energy Method According to the law of conservation of energy, if the energy loss of the system in the vibration process is neglected, for example, ignoring the damping effect, the sum of the potential energy and the kinetic energy of the system will remain a constant at any time. As in the static equilibrium position, the potential energy of the system is zero, so the system obtained the maximum kinetic energy at this time, recorded as Umaxo. Similarly, at the

Multi-Degree of Freedom System

151

moment that the system reaches the maximum displacement, because the speed is zero, the kinetic energy is zero, so all the energy switches into potential energy, recorded as Wmaxo. From the energy conservation theorem we know:

Wmax y x,t

U max

Y ( x )sin(

(4.63)

n

t

)

(4.64)

In which, Y(x) is vibration mode shape, ωn is frequency. The above equation is the partial derivative of time t, and the vibration velocity of the system is obtained. That is:

y x,t

Y ( x )cos

n

n

t

(4.65)

According to it, the kinetic energy of the system is obtained. That is:

U

1 2

l

1 2

m( x )y 2 x , t dx

0

2 n

cos 2

l nt

m( x )Y 2 ( x ) d x

0

(4.66) When cos(ωnt – φ) = 1, the system obtained the maximum kinetic energy.

U max

1 2

2 n

l

m( x )Y 2 ( x ) dx

0

m(x), y(

X l

Figure 4.11 Simplified calculation of simply supported beam.

(4.67)

152

Structural Dynamics

If only the bending deformation is consedered, the strain energy of the system is:

1 W 2

2

l

EI ( x )

y x ,t x2

0

2

2

d 2Y ( x ) dx EI ( x ) 0 dx 2 l

1 d x sin 2 2

n

t

(4.68) Similarly:

Wmax

d 2Y ( x ) EI ( x ) dx 2 0 l

1 2

2

dx

(4.69)

Substituting equation (4.67) and (4.69) into equation (4.63). It can obtained that: 2

d 2Y (x) EI ( x ) dx 2 0 l

2 n

l

dx (4.70)

2

m( x )Y ( x ) dx 0

In addition to the distribution mass m(x), if the system also possesses n concentrated masses (Figure 4.12). In this case, the kinetic energy of the system should not only include the kinetic energy of the distribution mass m(x) as shown in equation (4.66), but also the kinetic energy of concentrated mass, which is:

U

1 2

l

1 m( x ) y ( x , t )d x 2 0

n

2

m1

mi

mi y 2 ( xi , t )

(4.71)

i 1

mn x

y(xl,t) y xl l

Figure 4.12 Simplified calculation of the beam with concentrated mass.

Multi-Degree of Freedom System

153

So, the maximum kinetic energy of system is:

U max

1 2

l

1 m( x )Y ( x ) dx 2 0

2 n

2

n 2 n

miY 2 ( xi )

(4.72)

i 1

And the equation (4.70) can be rewritten as: l 2 n

0

EI x [Y ( x )]2 dx (4.73)

n

l

2

2

m( x )Y ( x ) dx 0

miY ( xi ) i 1

It can be noted that the natural frequency of the system can be easily acquired simply when the vibration form Y(x) of the system is known. If the displacement shape function Y(x) that has been assumed is exactly the same as the first principal mode, the approximation of the first frequency of the structure can be obtained. If it is close to the second mode, the approximation of the second order frequency can be obtained. But the problem is that the displacement function shape Y(x) is not known in advance. However, for the first frequency, according to some practical calculations, it can illustrate that as long as the assumed mode curve satisfies the boundary condition of the system and roughly conforms to the fundamental vibration form, the error of the natural frequency achieved by substituted which into the above equation will not be great. If the assumed mode function is accurate, the natural frequency obtained is also accurate. In general, it is difficult to accurately assume a high-order mode function, so Rayleigh (Rayleigh) is only used to calculate the basic frequency. Although the use of the Rayleigh method for high-order frequency effect is not good, for the symmetrical structure, it is easy to give its symmetrical mode and anti-symmetrical mode, the corresponding two natural frequencies can be obtained easily. In order to solve the first frequency, Rayleigh suggests that the static displacement generated by the self-weight load on the system m(x)g can be treated as the first mode Y(x) (note that if considering horizontal vibration, gravity should act in the horizontal direction.) So the work that it has done can be expressed as

W

1 2

l

m( x ) gY ( x ) dx 0

(4.74)

154

Structural Dynamics

Thus, for the structural systems with both distributed and concentrated mass, equation (4.73) can be rewritten as: n

l

m( x ) gY ( x ) dx 2 n

0

mi gY ( xi ) i 1 n

l

2

m( x )Y ( x ) dx 0

(4.75) 2

miY ( xi ) i 1

Of course, it is also possible to take the elastic expression of the structure under the action of a static load q(x) as the approximate expression of the mode curve, so that the above equation can be rewritten as: l

q( x )Y ( x ) dx 2 n

0

(4.76)

n

l

2

2

m( x )Y ( x ) dx 0

miY ( xi ) i 1

Example 4.4 Solve for the first frequency of the simply supported beam by energy method (EI is constant, the distribution mass is m). Solution: (1) Suppose that the first-order mode curve Y(x) of the simply supported beam is a parabola.

Y (x )

4ax (l x ) l2

when x=0, Y(0)=0; when x=l , Y(l)=0, It can be seen that this parabolic satisfies the boundary condition:

d 2Y dx 2

8a l2

Multi-Degree of Freedom System

155

According to equation (4.73): l

l

2

EI

EI ( x ) Y ( x ) dx 2 n

0

0 l

m( x )Y 2 ( x )dx

0

l

m 0

2

8a l2

dx 2

4ax (l x ) dx l2

64 EIa 2 / l 3 120 EI ml 4 8ma 2 l15

So, there is

n

10.95 EI m l2

(2) The deflection curve under uniformly distributed load q is taken as the mode shape curve Y(x):

Y (x )

q x 4 l 3 x 2lx 3 24 EI

Substituting it into equation (4.73) results in:

q2 (x 4 0 24 EI l

2 n

l

m 0

q x4 24 EI

l 3 x 2lx 3 ) dx

q 2 l 2 / 120 EI

2 3

3

n

9.87 EI m l2

l x 2lx

dx

So:

(3) Assume that the shape function is sinusoidal

Y ( x ) a sin

x l

q m 24 EI

2

31 9 l 630

156

Structural Dynamics

9.8696 EI m l2 It can be seen that since the sinusoidal curve is the exact solution of the first principal mode, the frequency ( n 9.8696 EI / m / l 2 ) obtained from it is the exact solution of the first frequency. In addition, the first two selected curves meet the boundary conditions almoatly, so the deviation of the results are minimal, and they are larger than the exact value. This is because the assumed curve is not the real vibration curve, which is equivalent to adding some extra constraints to the structural system. Then the stiffness of the system is increased, so the resulting frequency will be larger. Substituting it into equation (4.73) can similarly obtain

n

4.3.3 Ritz Method It can be seen from the previous example that the accuracy of the first frequency of the system obtained by the energy method depends on the accuracy of the assumed mode. Furthermore, only the upper limit of the fundamental frequency of the vibration can be acquired. In order to find the approximation of the higher order frequency and to make the lowest frequency close to the exact solution, the Rayleigh method was developed. The Ritz method is based on the Hamilton variation principle, which transforms the variation problem into the extreme value problem of multiple variable functions. Hamilton’s principle is a basic variation principle in analytical mechanics. It provides the criterion for judging the real and actual movements from all movements that are possible and satisfying the constraint situation. For the actual movement, the kinetic energy U, potential energy W and virtual work W of the elastic system must meet: t2

t2

W U dt t1

W dt

0

(4.77)

t1

For free vibration problem of the periodic (integral upper and lower limits take a cycle, that is 0 and 2π / ω), no damping structure of the system free vibration problem, the virtual work of above formula is zero. The Hamilton principle can be expressed as: in all possible motion state, the exact solution makes

Multi-Degree of Freedom System

157

2

W U dt

stationaryvalue

(4.78)

0

Substituting equation (4.66) and equation (4.68) into the above formula, because the time integral range is taken as a period, results in:

1 2

EI ]Y ( x )]2 dx

2 n

m( x )Y 2 ( x ) dx stationaryvallue

2

(4.79) Let Y(x) be the assumed mode curve and decompose it by the mode: n

Y ( x ) a1 y1 ( x ) a2 y 2 ( x )

an yn ( x )

ai yi ( x )

(4.80)

i 1

In which, y(x) are n independent displacement functions which satisfy the displacement boundary condition, a is undetermined parameter. Substituting the above equation into equation (4.79), the equation becomes:

1 EI a1 y1 ( x ) a2 y 2 ( x ) 2 2 n

2

ai yi ( x ) an yn ( x ) dx 2

m( x ) a1 y1 ( x ) a2 y 2 ( x )

2

ai yi ( x ) an yn ( x ) dx

n

1 2

n

a j ai EIyi ( x ) yi ( x )d x j ,i 1

2 n

a j ai m( x ) yi ( x ) y j ( x )d x i,j 1

(4.81) l

if kij

0

EIyi x y j ( x )dx , mij

l 0

m( x ) yi ( x ) y j ( x )dx , then

158

Structural Dynamics

1 2

n

n

kij i 1

2 n

mij ai a j

(4.82)

j 1

Using the stationary condition:

ai

0,(i 1, , n), then:

n 2 n

kij

mij a j

0,(i 1,

, n)

(4.83)

j 1

Writing it into matrix form:

[K ]

2 n

[ M ] {a} {0}

(4.84)

According to the law of Kramer, if the equations group has nonzero solutions (because the parameter ai are not all zero), the coefficient determinant should be equal to zero, thus 2 n

[K ]

[M] 0

(4.85) 2 n

Since the above equation is an n-th algebraic equation about the approximation of the first n natural frequencies can be obtained.

, an

Example 4.5 Solve out the natural frequency of the equal section cantilever beam (shown in Figure 4.13) by using the Ritz method. m,EI x l y

Figure 4.13 Cantilever beam structure.

Solution: Assume that the approximate mode function is:

x Y ( x ) a1 1 l

2

x x a2 1 l l

2

Multi-Degree of Freedom System

159

By equation (4-81), the constant kij and mi can be obtained. As shown in the following:

[K ]

4 EI l3 2EI l3

2EI l3 4 EI l3

ml 5 ml 30

[M]

,

ml 30 ml 105

Therefore, the frequency equation is:

4 EI l3 2 EI l3

2 n

ml 5

2 n

ml 30

2 EI l3 4 EI l3

2 n

2 n

ml 30 ml 105

0

Namely,

0.794m 2l 2

4 n

972 EIm

2 n

/ l 2 12000( EI / l 3 )2

0

the roots of the equation are

n1

3.55

n1

34.81

EI EI (The exact value was 3.516 ), 4 ml ml 4 EI EI (The exact value was 22.035 ) 4 ml ml 4

In order to improve the computational accuracy of ωn2, the following four functions are available.

Y ( x ) a1 y1 ( x ) a2 y 2 ( x ) a3 y3 ( x ) a4 y 4 ( x ) In which, the first two terms y1(x), y2(x) are in compliance with equation (a), and y3(x), y4(x) are respectively expressed:

160

Structural Dynamics

y3 ( x )

x l

y4 (x )

x l

0.5

x l

0.75

x 1 l

x l

0.5

2

x l

x 1 l

2

The first two order frequencies of the structure can be obtained as:

n1

3.516

EI , ml 4

n2

22.159

EI ml 4

It can be revealed that it would be better to take more items in progression of hypothetical modes if a more accurate value needs to be obtained.

4.3.4 Matrix Iteration Method The matrix iteration method is also called the Stodola method or power method, this method calculates the frequency and mode of the structure by the gradual approximation method, which can be used to solve out the first few modes and frequencies of the structure. As mentioned earlier, the free vibration equation (4.20) of a multi-degree of freedom system can be expressed as:

[K ]{ X } Multiplying the above equation by

1 2 n

Let

1 2 n

2 n

[ M ]{ X }

1 2 n

(4.86)

[K ] 1 gives:

{ X } [K ] 1[ M ]{ X }

(4.87)

,[ ] [K ] 1[ M ], the above equation becomes:

{ X } [ ]{ X }

(4.88)

Multi-Degree of Freedom System

161

Now assuming that {X}0 is the first approximation of the first mode, and performing regularization (which suppose that the amplitude of one particle is 1, usually the amplitude of the first or the n-thparticle), substitutes it into the left side of the above equation and makes:

{ X }1 [ ]{ X }0

(4.89)

If {X}0 is the real solution of the first mode, it must be:

{ X }1

{ X }0

(4.90)

If the above equation is invalid, again let:

{ X }0

{ X }1

(4.91)

This iteration process is repeated until the two adjacent iterative results are similar. It should be pointed out that the physical meaning expressed by the column vector of the mode is the relative displacement between the particles. {X} is not the absolute value but the relative value, because of this, the mode {X} should be performed normalize before each iteration so that it is convenient to compare between the two modes before and after the iteration. It is also more efficient to obtain the true value. Example 4.6 A system with three degrees of freedom, its mass matrix [M] and stiffness matrix [K] is respectively:

[M]

2250 0 0 2540

0 t, 560

[K ]

14.46 9.03 9.03 17.26 0 8.23

0 8.23 8.23

105 kN/m,

Solve out the first-order frequency and mode of the structure by matrix iteration method.

162

Structural Dynamics

Solution: Because of

K

0.1842 0.1842 0.1842

1

0.1842 0.2949 0.2949

0.1842 0.2949 0.4164

10 5 m/kN,

There is

469.6133 467.7716 103.1308 469.6133 749.0562 165..1463 469.6133 749.0562 233.1900

[ ]

10

5

Assuming {X}0 = {1 1 1}T, substituting it into equation (4.89), there are

1

{X }

469.6133 467.7716 103.1308 469.6133 749.0562 1665.1463 469.6133 749.0562 233.1900 1 1 1

1.0405 1.3838 1.4519

10

10

5

2

After performing regularization of shape, there is: 0 1 1 X (1) , the X (1) {0.717 0.953 1.000}T here, now because of X second iteration is needed to be performed. 0 {0.717 0.953 1.000}T , and then At the second iteration, first let X substituting the equation (4.89) into it:

X

1

469.6133 467.7716 103.1308 469.6133 749.0562 165.1463 469.6133 749.0562 233.1900 0.717 0.953 1.000

0.8855 1.2156 1.2837

10

2

10

5

Multi-Degree of Freedom System 1

X ( 2 ) {0.6898 0.9470 1.0000}T ,  repeatafter the third iteration, there is

After the normalization, ing the above process, X ( X

1 (3) 1 (2)

{0.6870 0.9462 1.0000}T , X

1

163

the

result

is

very

approximate

), so the approximate solution of the first-order model is:

(3)

X

0.6870 0.9462 1.0000

1

T

So from equation (4.87), there is:

469.6133 467.7716 103.1308 469.6133 749.0562 165.1463 469.6133 749.00562 233.1900

0.6870 0.9462 1.0000

1 2 n

10

0.6870 0.9462 1.0000

5

Noting that the above equation is actually three independent equations, it can be solved according to any type of solution, such as by type 3:

1 2 n

1.0(469.6133 0.6870 749.0562 0.9462 233.1900 1.0) 10

5

so ωn1 = 8.89rad/s Then it will prove that the frequency and mode obtained by the above-mentioned iterative method are the first frequency fundamental frequency and the corresponding mode. From the equation1 (4.89) we can see that after two times iterations, the mode vector, X ( 2 ) (the subscript indicates the number of iterations), is:

X

1

X

(2)

1

X

(1)

2

0

Similarly, after k times iterations the equation becomes:

X

1 (k )

k 1

X

0

k

X

0

X

0

164

Structural Dynamics

Since the assumed {X}0 can be expressed as a linear combination of the true mode vector {Xi} of the system:

X

0

a1 X1

a2 X 2

an Xn

In which, ai is a constant, {Xi} is the i-th mode of the system. So, according to [β]{Xi} = αi{Xi} (i represents the i-th mode, resulting in:

X

0

a1

X1

Xn a1a1 X1

a2

X2

a2a2 X 2

1 i

2 ni

),

an anan Xn

Similarly: k

X

0

a1k a1 X1

a2k a2 X 2

ank an Xn

Dividing both ends of the above equation by

1 k 1

k 1

becomes: k

k k

[ ] X

0

2

a1 X1

n

a2 X 2

1

an Xn

1

Because of 0 < ω1 < ω2 α2 > … >αn > 0, which means that when k is large enough, (αi / α1)k ≈ 0, therefore:

X

1 (k )

k

X

0

k 1 1

a X1

It can be seen that after k times iterations, the difference between the [β]k{X}0 and the exact value {X1} of the first mode is only a constant term 1k a1, and the normalization done after each iteration can eliminate the influence of this constant term. So, after finishing many repeated iterations:

X

1 (k )

X1

Multi-Degree of Freedom System

165

The mode iteration method can not only solve the fundamental frequency and first order mode of the system, but also can be used to determine the high frequency and the corresponding vibration mode. However, in order to do this, it is necessary to deal with the assumed iteration vector appropriately. Which means that in order to decide the second mode, the effect of the first mode must be eliminated in the assumed iteration vector. When the third mode is determined, the effects of the first and second modes should eliminate the effects of the first and second modes, and so on. Based on this law, before performing the iteration, let a1 = 0, the final result of the iteration will converge to the second mode; Let a1 = a2 = 0, which will converge to the third mode, and so on. Therefore, the concrete way to obtain the system i-order frequency or mode by iteration method is as follows: Premultipling the linear combination of the mode vector {Xi} by {Xi}T [M], and using the orthogonality of the mode results in:

Xi

T

M X

0

a1 Xi

T

T

an Xi a1 Xi

[ M ] X1

T

ai Xi

T

[ M ] Xi

[ M ] Xn

[ M ] Xi

So:

Xi

ai

Xi

T T

[M] X

0

[ M ] Xi

In order to eliminate the preceding r-order mode component in the r

0

assumed mode, the initial iteration vector is taken as X substituting aj into it becomes: r

X

0

r

ai X j

X

0

j 1

Xj

j 1 r

In which, the S

Xj

r

E j 1

Xj Xj

Xj

Xj T

T

T T

[M]

[M ] X j

[M ] X [M] X j

aj X j , j 1

0

S

r

X

0

is the clearing type matrix.

166

Structural Dynamics

In the actual iteration process, in order to avoid that the vibration may contain the former r-order mode components, it is required that which should be premultiplied by the matrix has been cleared each time. In other words, when the first frequency of the system is obtained by the matrix iteration method, the process can be carried out according to equations (4.88) to (4.91). For solving the higher order frequencies, replacing [β] in the formula (4.88), then it can be obtained by performing the matrix iteration. Example 4.7 Solve out of 2nd shape and frequency of example 4.6 Solution: Because that {X1},[M],[K],[β],α1 have been known:

X1

T

2550 0 0 [ M ]{ X1 } {0.6870 0.9462 1.00} 0 2540 0 0 0 560 0.6870 0.9462 1.0000

4037.60

2550 0 0 0.6870 { . 0 0 . 9462 1 . 00 } 0 6870 0.9462 0 2540 0 0 0 560 1.0000

{ X1 }{ X1 }T [ M ]

1.2035 1.6511 0.3847 1.6576 2.2740 0.5299 1.7519 2.4033 0.5600 1 [S]1

1 1

103

1.2035 1.6511 0.3847 1.6576 2.27440 0.5299 1.7519 2.4033 0.5600

0.7019 -0.4089 -0.0953 -0.4105 0.4368 -0.1312 -0.4339 -0.5952 0.8613

103 / 4037.660

Multi-Degree of Freedom System

[ ]1 [ ][S]1

469.6133 467.7716 103.1308 469.6133 749.0562 165.1463 469.6133 749.0562 233.1900

10

167

5

0.7019 -0.4089 -0.0953 -0.4105 0.4368 -0.1312 -0.4339 -0.5952 0.8613 0.9284 -0.4912 -0.1731 -0.4954 0.3683 -0.0081 -0.7907 -0.0368 0.5780

10

3

Assuming that the initial approximation of the second-order mode is {1.00 0.00 1.00}T , executing the iteration by the matrix iteration method of Example 4.6, in which [β] takes [β]1. So the resultant is { X 2 } {0.98 0.49 1.00}T , 2 27.20rad/s .

{ X 2 }0

4.3.5 Subspace Iteration Method The subspace iteration method is also called the parallel iteration method, which is essentially a repeated use of the Ritz method and the matrix iteration method for a set of test vectors. Matrix iterations can only find one eigenvalue and eigenvector of matrix by one iteration, but subspace iteration can find the first few largest eigenvalues and eigenvectors of matrix by one iteration. The basic principle is that the original n linearly independent eigenvectors (modes) {X}1, {X}2, …,{X}n of the matrix [X] compose an n-dimensional vector space. The p (p