Elements of Structural Dynamics E-Book

Debasish Roy

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 602

Veröffentlichungsjahr: 2012

Beschreibung

Structural dynamics is a subset of structural analysis which covers the behavior of structures subjected to dynamic loading. The subject has seen rapid growth and also change in how the basic concepts can be interpreted. For instance, the classical notions of discretizing the operator of a dynamic structural model have given way to a set-theoretic, function-space based framework, which is more conducive to implementation with a computer. This modern perspective, as adopted in this book, is also helpful in putting together the various tools and ideas in a more integrated style. 

Elements of Structural Dynamics: A New Perspective is devoted to covering the basic concepts in linear structural dynamics, whilst emphasizing their mathematical moorings and the associated computational aspects that make their implementation in software possible. 

Key features:

• Employs a novel ‘top down’ approach to structural dynamics.
• Contains an insightful treatment of the  computational aspects, including the finite element method, that translate into numerical solutions of the dynamic equations of motion.
• Consistently touches upon the modern mathematical basis for the theories and approximations involved.

Elements of Structural Dynamics: A New Perspective is a holistic treatise on structural dynamics and is an ideal textbook for senior undergraduate and graduate students in Mechanical, Aerospace and Civil engineering departments. This book also forms a useful reference for researchers and engineers in industry.

Leseprobe

Table of Contents

Title Page

Copyright

Preface

How To Read This Book

Acknowledgements

Introduction

General Notations

Chapter 1: Structural Dynamics and Mathematical Modelling

1.1 Introduction

1.2 System of Rigid Bodies and Dynamic Equations of Motion

1.3 Continuous Dynamical Systems and Equations of Motion from Hamilton's Principle

1.4 Dynamic Equilibrium Equations from Newton's Force Balance

1.5 Equations of Motion by Reynolds Transport Theorem

1.6 Conclusions

1.7 Exercises

Notations

References

Chapter 2: Continuous Systems – PDEs and Solution

2.1 Introduction

2.2 Some Continuous Systems and PDEs

2.3 PDEs and General Solution

2.4 Solution to Linear Homogeneous PDEs – Method of Separation of Variables

2.5 Orthonormal Basis and Eigenfunction Expansion

2.6 Solutions of Inhomogeneous PDEs by Eigenfunction-Expansion Method

2.7 Solutions of Inhomogeneous PDEs by Green's Function Method

2.8 Solution of PDEs with Inhomogeneous Boundary Conditions

2.9 Solution to Nonself-adjoint Continuous Systems

2.10 Conclusions

Exercises

Notations

Chapter 3: Classical Methods for Solving the Equations of Motion

3.1 Introduction

3.2 Rayleigh–Ritz Method

3.3 Weighted Residuals Method

3.4 Conclusions

Exercises

Notations

Chapter 4: Finite Element Method and Structural Dynamics

4.1 Introduction

4.2 Weak Formulation of PDEs

4.3 Element-Wise Representation of the Weak Form and the FEM

4.4 Application of the FEM to 2D Problems

4.5 Higher Order Polynomial Basis Functions

4.6 Some Computational Issues in FEM

4.7 FEM and Error Estimates

4.8 Conclusions

Exercises

Notations

Chapter 5: MDOF Systems and Eigenvalue Problems

5.1 Introduction

5.2 Discrete Systems through a Lumped Parameter Approach

5.3 Coupled Linear ODEs and the Linear Differential Operator

5.4 Coupled Linear ODEs and Eigensolution

5.5 First Order Equations and Uncoupling

5.6 First Order versus Second Order ODE and Eigensolutions

5.7 MDOF Systems and Modal Dynamics

5.8 Damped MDOF Systems

5.9 Conclusions

Exercises

Notations

Chapter 6: Structures under Support Excitations

6.1 Introduction

6.2 Continuous Systems and Base Excitations

6.3 MDOF Systems under Support Excitation

6.4 SDOF Systems under Base Excitation

6.5 Support Excitation and Response Spectra

6.6 Structures under Multi-Support Excitation

6.7 Conclusions

6.8 Exercises

6.9 Notations

Chapter 7: Eigensolution Procedures

7.1 Introduction

7.2 Power and Inverse Iteration Methods and Eigensolutions

7.3 Jacobi, Householder, QR Transformation Methods and Eigensolutions

7.4 Subspace Iteration

7.5 Lanczos Transformation Method

7.6 Systems with Unsymmetric Matrices

7.7 Dynamic Condensation and Eigensolution

7.8 Conclusions

Exercises

Notations

Chapter 8: Direct Integration Methods

8.1 Introduction

8.2 Forward and Backward Euler Methods

8.3 Central Difference Method

8.4 Newmark-β Method – a Single-Step Implicit Method

8.5 HHT-α and Generalized-α Methods

8.6 Conclusions

Exercises

Notations

Chapter 9: Stochastic Structural Dynamics

9.1 Introduction

9.2 Probability Theory and Basic Concepts

9.3 Random Variables

9.4 Conditional Probability, Independence and Conditional Expectation

9.5 Some oft-Used Probability Distributions

9.6 Stochastic Processes

9.7 Stochastic Dynamics of Linear Structural Systems

9.8 An Introduction to Ito Calculus

9.9 Conclusions

Exercises

Notations

References

Appendix A

Appendix B

Appendix C

Appendix D

Appendix E

Appendix F

Appendix G

Appendix H

Appendix I

Index

This edition first published 2012

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Library of Congress Cataloging-in-Publication Data

Roy, Debasish Kumar, 1946-

Elements of structural dynamics : a new perspective / Debasish Roy, G Visweswara Rao.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-118-33962-6 (hardback)

1. Structural dynamics. I. Gorti, Visweswara Rao. II. Title.

TA654.R69 2012

624.1′71-dc23

2012011742

A catalogue record for this book is available from the British Library.

Print ISBN: 9781118339626

Preface

To the best of the authors' knowledge, this book is one of the few attempts at a top-down approach to the subject of structural dynamics. Thus, unlike the oft-treaded route followed in most texts, we depart from introducing the single-degree-of-freedom (SDOF) oscillator and its response features in the first chapter and rather start with the basic principles of linear momentum balance of isotropic and linearly elastic systems, which in turn yield the governing equations of motion in structural dynamics. Whilst an SDOF oscillator is commonly viewed as, and rightly so, as one of the simplest building blocks for the instruments of mathematical modelling needed for an insightful understanding of the subject, it is generally far removed (at least from a functional perspective) from what we usually qualify as structural dynamic systems of engineering interest. Accordingly, despite its advantages in an appropriately sequenced bottom-up exposition, the SDOF system hardly corresponds to what we often perceive and loosely describe as ‘real-life’ systems. This oft-practiced mode of instruction could leave an inquisitive student somewhat flummoxed until about the last chapter, where he finally gets to connect things up and realise the importance of what he studied in the preceding chapters. Our motivation in writing this treatise has mainly been to upend this scheme of instruction and fashion the discourse that starts directly with an attempt to mathematically model and computationally treat the ‘real-life’ structural systems, whilst not sacrificing the important feature of a suitably graded exposition with gradually increasing complexity. It is from this perspective that the book can hopefully lay some claim to novelty, mainly aimed at holding the interest of a student right from the start and all through afterwards. In particular, we provide below an overview of the contents in the form of a chapter-wise breakup and include, at appropriate places, some suggestions as to what could be covered within an introductory course covering a semester at an advanced under-graduate or an early graduate level.

The book begins with a prologue providing a few introductory remarks on the significance of ‘dynamic’ analyses of structural systems. The necessity to have valid mathematical models for both the external loadings and systems is highlighted. Chapter 1 lays down the principles underlying the mathematical models describing the dynamic equations of motion for continuous structural systems. The reader is familiarised with the resulting partial differential equations (PDEs) derived via different routes viz., the familiar Hamilton's principle involving energy expressions, Newton's force balance yielding the state of stress inside a continuum from a solid mechanics point of view and Reynolds transport theorem using the laws of conservation of mass and linear momentum balance. Chapter 2 brings into focus the insight one derives from the knowledge of an assemblage of analytical methods for solving linear PDEs governing the dynamics of a class of continuous systems. Concepts on free vibration solutions based on an eigenanalysis are introduced in the chapter for both self-adjoint and non self-adjoint systems. The elementary notion of Fourier series expansion of a function in a Hilbert space setting is formalised. This is followed by the use of such expansions, based on eigenfunctions, to solve a few dynamic systems governed by inhomogeneous PDEs with both homogeneous and inhomogeneous boundary conditions. The expansion forms a basis for mathematically breaking up a continuous system into a sequence of SDOF oscillators that are more easily inverted. Given the general intractability of analytical approaches for solving PDEs of motion of complex (large) systems, Chapter 3 describes approximate projection methods that attempt to solve the equations via either the extremization of a functional or orthogonalization of certain residuals. Rayleigh–Ritz method and weighted residual methods like Galerkin, falling under this class of approximate methods, are covered in this chapter. An exposition of these classical methods brings into relief the notion of semi-discretization of a continuum mathematical model leading to its discrete, finite-dimensional counterpart (with finite degrees of freedom, usually known as multi-degree of freedom (MDOF) systems). A piecewise (i.e. spatially localised) application of these methods sets the stage for a brief, yet very relevant, introduction to the finite element method (FEM), which forms the subject of Chapter 4.

Analysis of MDOF systems governed by coupled ordinary differential equations (ODEs) that in turn result from the semi-discretization of a continuous system (by either a classical method or the FEM) is dealt with in Chapter 5, with particular attention to solutions of linear time invariant (LTI) systems. Eigenvalue analysis, being the prime mover to obtaining free and forced vibration solutions of linear undamped or damped MDOF systems, is described in some detail in this chapter. In view of their practical importance in the seismic qualifications of structures/equipments, Chapter 6 is devoted to the problem of structural dynamic analysis under support excitations in both time and frequency domains. Here again, the top-down approach of treating a continuous structure first, only to move to the discrete MDOF model later and to the simplest building block of an SDOF oscillator at the last stage, is followed in order to be in sync with the overall expositional format of this book. The response spectrum method vis-à-vis time history methods and a practitioner's preference of the former for design purposes are also briefly touched upon.

Like Chapter 7, Chapter 8 has again a tilt to the broad area of computational structural dynamics in that it is exclusively devoted to a family of eigenvalue extraction techniques for the generalized eigenvalue problem, starting from simple power and inverse iteration methods to the powerful Lanczos method. Use of shifting for better convergence and the role of multiple eigenvalues in obtaining an eigensolution are discussed. Methods suitable for both symmetric and unsymmetric systems are covered along with illustrative examples. Whilst the fundamental aspects of these techniques are always highlighted, the issue of their numerical implementation in the context of large systems is also not lost sight of. Consistent with this approach, algorithms are provided for each of the eigensolution techniques so as to facilitate their implementations on a computer. Direct integration schemes to solve coupled ODEs represented by MDOF systems are presented in Chapter 9 with specific applications to linear structural dynamic systems. The necessary attributes of an efficient integration scheme – consistency, stability and convergence and numerical dissipation are discussed at length in this chapter along with supportive examples.

Chapter 10 briefly introduces what is somewhat loosely referred to as stochastic structural dynamics (or random vibration), which provides a useful tool in propagating the input (forcing) uncertainty, as in earthquake or wind induced structural vibrations, through the equations of motion to characterise the response uncertainty. In laying out the methodology, we have here emphasised a modern and insightful treatment of the subject based on the elements of stochastic calculus (Ito calculus in particular).

How To Read This Book

If used as a textbook for a one-semester course of Structural Dynamics at the undergraduate level, it is possible to start with and cover Chapter 11 in full, followed by parts of Chapter 12 (for instance, a few 1D examples from Sections 2.2–2.4 restricted to only simply supported beam equations as examples; Sections 2.5 and 2.6). Whilst Chapter 3 may be entirely covered, Chapter 4 on the FEM may only be used to introduce the basic notion of semi-discretization (again emphasising 1D examples and without covering the error estimates). This may be followed by a nearly full coverage of Chapters 5 and 6, whilst only touching upon the basic eigensolution approaches in Chapter 7 (e.g. Section 7.2 in full and the Jacobi method as well as its convergence in). Finally, Chapter 8 may be barely touched to introduce only the Euler and Newmark methods. In case of a one-semester long post-graduate course, the above scheme may be generally followed (possibly with the addition of examples on Timoshenko beam and 2D plate bending in Chapter 2, finite element error estimates in Chapter 4) and Chapter 9 may also be used towards the end to introduce the notion of stochastic processes and consequently the fundamental aspects of Ito calculus in evaluating the response moments.

However, if used as a reference, a reader (depending on the background) may choose to go through the chapters in this book in a selective, non-sequential manner to gain additional insights into the subject.

Acknowledgements

Writing this book has taken some effort spanned over a time interval of about one and half years, a significant part of which should have deservedly gone to our family members to whom we express our gratefulness not entirely unmixed with a sense of apology.

Introduction

Dynamics is an inherent nature of a system reacting to its environmental forces. The dynamic response may be described by some quantitative measures of the system behaviour (e.g. displacements, stresses, etc.) evolving with time. As time progresses, the transient phenomena may disappear and it is possible that the system attains a steady state. The transients disappear typically due to the internal damping present in structural dynamic systems. The steady state may be independent of time in which case the system is said to have reached the static equilibrium position. For example, a bridge/ship deck under an impact loading may undergo oscillations and eventually reach a steady state. This may also be the case with such time-invariant (static) loads acting on the system forever, once the initial perturbations die down. Certain time-varying loads of sustained nature may be periodic in time so that the system dynamically responds with (some of) the response measures also exhibiting periodicity, especially following some initial cycles of transients. For example, a rotating system such as a generator or a pump motor in a power plant experiences such periodic loads due to the inherent rotor mass imbalances. This may in turn cause excessive vibrations in the adjacent structural components/equipments. One needs to take care of the resulting response magnitudes of both the primary/secondary systems that may exceed the safe allowable limits. On the other hand, loads such as those caused by seismic events or bomb blasts may be of characteristically shorter duration. If the systems need to be designed to withstand such shock loads without a major failure, the interest for an analyst lies in mitigating or damping out the vibration levels, if not suppressing them altogether.

All the environmental loads for a structural system may not be as easily amenable for mathematical representation, which, however, constitutes a pre-requisite for a subsequent analysis to be carried out on the system. One needs to use the right blend of skill and experience in (approximately) modelling the environmental loads. The exercise is all the more fraught with difficult in that environmental loads are inherently of uncertain characteristics, that is they are often not modelled deterministically. In particular, some of them may admit modelling through what are referred to as stochastic processes. For example, wind, wave or earthquake excitations could be interpreted as instances of a stochastic process (Crandall and Mark, 1963; Lin, 1967). Stochastic processes must be treated within the framework of probability theory (Papoulis, 1991), with random variables as the basic building block. In any case, the subject of structural dynamics is relevant and may lay its claim to some level of completeness only if the excitations are properly modelled.

From the viewpoint of analysis of a structural system – a building, or a mechanical component such as a crank shaft or a space structure – it is imperative that we reduce the system into a mathematically (and computationally) viable form. This in turn requires efficient tools, say to decompose the generally complex system on hand into components/elements whose mathematical models are individually far easier to handle and analyse. The mathematical models of a truss, an Euler–Bernoulli beam, a shear beam, a plate a shell or a solid body as a continuum (which typically form system components) are fortunately derivable from Newton's principle of momentum balance combined with the constitutive equations relating stresses and strains. Despite the apparent complexity with a structural system in its original form, it often admits decomposition into such simpler elemental sub-systems. A skillful approach is nevertheless warranted in carving out an acceptable model comprising of different types of these known elemental systems and finally realise reasonable results. In many cases, it is a trade-off between the effort put in the modelling and the availability of computational facilities to treat the model. With the advent of high capacity personal computers/workstations, it is no wonder that the modelling ability has of late grown manifold with the result that one is now enabled to model and solve for the response of complex structural dynamic systems with lesser effort. However, a need is always present for an adequately informed, yet intuitive, approach whilst arriving at the mathematical models.

Shown below, by way of an example, is a possible mathematical model of a turbo-generator foundation (Figure P1) that carries the turbine generator set and other auxiliary structural components. Forming this model, by all means, seems to be a formidable task. Figure P2 shows yet another case of what could possibly be labelled a ‘complex structural system’ – a high voltage transmission line tower with numerous bolted steel angle members. A major difficulty in the modelling exercise is to resolve the question on what constitutes a fine enough discretization to arrive at converged numerical solution for a structural system with continuous mass and stiffness distributions. It also possible that the solution corresponding to too fine a discretization may have high numerical pollution. An appropriate reduction of a continuous, infinite-dimensional system to a finite dimensional one (model reduction/semi-discretization) is again an issue that is mostly resolved through professional experience with such problems and the scientific information gleaned from this book alone might prove quite inadequate to meet this challenge. Nevertheless, only professional experience, not assisted by a suitable scientific basis, is likely to spring nasty surprises especially in somewhat uncharted territories. For instance, it is readily possible to conceive of a structural system which, whilst yielding strictly positive definite stiffness matrices, may not admit a unique solution under a certain class of boundary conditions. No amount of jugglery with discretization, based on just professional skills, is going to make the computed solution unique in these scenarios.

Figure I.1 Turbo-generator foundation (Indian Standard 2974 (Part 3): 1992); finite element model with 8-noded solid elements (Courtesy of Cranes Software International Ltd., Bangalore, India)

Figure I.2 Transmission line tower with two-noded truss elements (Indian Standard 802 (part 1/ Sec 1): 1995; finite element model (Courtesy of Cranes Software International Ltd., Bangalore, India)

Whilst it may not be the only available method to do so, the finite element method (FEM) (Zienkiewicz and Cheung, 1967; Hughes, 1987; Cook, Malkus and Piesha, 1989) is nevertheless the most widely used route to accomplish such model reduction. It is basically a discretization tool with a wide range of simple elemental systems from which one can make an informed choice in building up the finite dimensional mathematical model of a system on hand.

In any case, the subject of structural dynamics which, in recent times, has encompassed diverse fields of interest – from structural system identification, health assessment or monitoring, control, nanostructures and opto-electro-structural systems – is too wide in scope to be covered in a single volume, especially one such as this dealing only with the elementary building blocks of the subject. Specifically, this book is devoted to linear structural dynamics with emphasis only on the basic concepts of vibration theory under deterministic and stochastic excitations, whilst not losing sight of the recent links the subject has with computational mechanics aided by such discretization strategies as the FEM. For an analyst or a designer, a structural system or component is often a 3D continuum, which could possibly admit a dimensional reduction to 2D or even 1D continuum based on some ‘engineering’ (as against mathematically rigorous) approximations. In our setting, the mathematical model representing the vibratory state of a continuous system is often a partial differential equation (PDE). Deriving these equations of motion from the basic principles of continuum mechanics and solving them even for simple systems provide a great insight into some of the generic aspects of the system behaviour. It also helps in understanding the need for discretization and the philosophy behind the construction of discrete models, especially for complex structural systems which are otherwise not amenable to an analytical solutions.

References

Cook, R.D., Malkus, D.S. and Piesha, M.E. (1989) Concepts and Applications of Finite Element Analysis, 3rd edn, John Wiley & Sons, Inc., New York.

Crandall, S.H. and Mark, W.D. (1963) Random Vibration in Mechanical Systems, Academic, New York.

Hughes, T.R.J. (1987) The Finite Element method, Prentice Hall International, Inc.

Lin, Y.K. (1967) Probabilistic Theory of Structural Dynamics, McGraw-Hill, New York.

Papoulis, A. (1991) Probability, Random Variables and Stochastic Processes, 3rd edn, McGraw-Hill, Inc.

Zienkiewicz, O.C. and Cheung, Y.K. (1967) The Finite Element Method in Structural and Continuum Mechanics, McGraw-Hill, London.

General Notations

A

Area of cross-section

A

Matrix operator (linear transformation)

B

Partial differential operator in strain-displacement relationship

c

Damping coefficient in an SDOF oscillator

c

Matrix of the elasticity constants in stress-strain constitutive relationship

C

Damping matrix

The field of complex numbers

Functions that, together with their first

m

derivatives, are continuous on (

a,b

)

E

Young's modulus

Eigenspace corresponding to the eigenvalue λ

External forcing functions

F

Deformation gradient

Body force

Surface force

Impulse response function of an oscillator

Impulse response function of mode

Complex response function (Fourier transform of )

I

Moment of inertia

I

Energy functional, identity matrix

J

Determinant of

F

k

Stiffness of an SDOF oscillator

K

Stiffness matrix

l

Length parameter

L

Lagrangian,

L

Differential operator

m

Lumped mass in an SDOF oscillator, mass density per unit length for 1D systems, mass density per unit area for 2D systems

M

Mass matrix

N

Size of the matrix

N

Null space operator

The natural numbers

Vector of generalised coordinates

Set of real numbers (real line)

Non-negative real numbers

n

-Dimensional Euclidean space

Matrices with real elements

t

Time variable

T

Kinetic energy

Approximate solution

Displacement field variable

Unit-step function

V

Potential energy

x

Coordinate axes

X

Coordinate axis

Y

Coordinate axis

Z

Coordinate axis

Z

The integers

Dirac delta function (see Equation 2.180)

Kronecker delta (see Equation 2.167)

Boundary domain

Strain tensor

Strain components

Eigenvalues

Eigenpair of eigenvalue and associated eigenvector

Frequency (eigenvalue) parameter

v

Poisson's ratio

Damping ratio of an SDOF oscillator

Mass density

Rayleigh quotient

Stress tensor

Stress components

Radian frequency

Natural frequency of an SDOF oscillator

Damped natural frequency of an SDOF oscillator

Laplacian operator:

Orthonormal eigenfunctions of (for a continuous system)

Matrix of eigenvectors (

)

Diagonal matrix containing eigenvalues on the diagonal

Material domain

Laplace transform

Gradient of a function:

Norm

Chapter 1

Structural Dynamics and Mathematical Modelling

1.1 Introduction

Structural dynamics finds wide application in all areas of engineering – civil, mechanical, aerospace, marine and many others. Excellent text books and treatises initiating a newcomer to structural dynamics are available in Meirovitch (1967), Craig (1981), Clough and Penzien (1982). For an academic learner and a practising engineer, it is particularly important to assimilate the basic concepts, judiciously apply them whilst assessing or predicting the performance of the structure and interpret the results in a coherent and meaningful fashion. The material presented in this chapter is mainly aimed at serving this purpose. The mathematical rigour in the presentation is thus kept at a level consistent with the above aim. The effort has been to emphasise the basic concepts whilst keeping in mind their usefulness from a practical perspective.

Dynamic equations of motion are derivable based on variants of Newton's principle of force balance and they typically appear in the form of partial differential equations (PDEs). Any structural system with varying stiffness and mass distribution over its volume may be (mathematically) modelled as a continuum having, in principle, infinite degrees of freedom (dof s). The degrees of freedom generally refer to the unknown displacements at any spatial (material) point in the system. For a solid body modelled as a continuum, there exist 3 dof s – for example three orthogonal translations – at a material point. However, for certain cases wherein the measure (length or span) of the material domain along one or two spatial dimensions is much less compared to the other(s), the mathematical model can be considerably simplified through a dimensional descent, that is by appropriately averaging over such dimensions leaving only the dominant dimension(s) in the reduced model. This is, for instance, the case with a plate (or a beam), which is derivable from the 3D continuum model through such a descent by averaging across the so-called thickness dimension(s). For such dimensionally reduced continuum models, an additional set of three rotation degrees of freedom at every material point needs to be introduced in order to capture the post-deformed structural orientations across the eliminated dimensions. The governing PDEs of motion (in the retained space variables and time) are typically derived based on the principles of virtual work and variational calculus (Hughes, 1987; Reddy, 1984, Humar, 2002).

In practice, further simplification is achieved by reducing the governing PDEs of motion into a system of ordinary differential equations (ODEs) through a process called discretization, wherein the spatial variable(s) in the governing equations is (are) removed, leaving time as the only independent variable. The process of discretization typically involves expressing the dependent response function in a (piecewise smooth) series involving known functions of the space variable(s) (often referred to as the shape functions) and unknown functions of time (often called generalized coordinates). This finally yields, upon some form of averaging or integration over the space variable(s), a system of ODEs of motion. The finite element method (FEM) is probably the most widely employed tool for such discretization based on spatially localised and piecewise smooth shape functions. Pending further elaboration of these ideas on discretization, some of which will be taken up subsequently in this book, we presently focus in this chapter on the basic principles of deriving the equations of motion based on virtual work.

1.2 System of Rigid Bodies and Dynamic Equations of Motion

To start with, it is instructive to consider a few simple cases of rigid-body dynamics, wherein the governing equations are in the form of ODEs.