Analytical and Numerical Methods for Vibration Analyses E-Book

Jong-Shyong Wu

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

Seitenzahl: 698

Veröffentlichungsjahr: 2013

Beschreibung

Illustrates theories and associated mathematical expressions with numerical examples using various methods, leading to exact solutions, more accurate results, and more computationally efficient techniques

This book presents the derivations of the equations of motion for all structure foundations using either the continuous model or the discrete model. This mathematical display is a strong feature of the book as it helps to explain in full detail how calculations are reached and interpreted. In addition to the simple 'uniform' and 'straight' beams, the book introduces solution techniques for the complicated ‘non uniform’ beams (including linear or non-linear tapered beams), and curved beams. Most of the beams are analyzed by taking account of the effects of shear deformation and rotary inertia of the beams themselves as well as the eccentricities and mass moments of inertia of the attachments.

• Demonstrates approaches which dramatically cut CPU times to a fraction of conventional FEM
• Presents "mode shapes" in addition to natural frequencies, which are critical for designers
• Gives detailed derivations for continuous and discrete model equations of motions
• Summarizes the analytical and numerical methods for the natural frequencies, mode shapes, and time histories of
• straight structures
• rods
• shafts
• Euler beams
• strings
• Timoshenko beams
• membranes/thin plates
• Conical rods and shafts
• Tapered beams
• Curved beams
• Has applications for students taking courses including vibration mechanics, dynamics of structures, and finite element analyses of structures, the transfer matrix method, and Jacobi method

This book is ideal for graduate students in mechanical, civil, marine, aeronautical engineering courses as well as advanced undergraduates with a background in General Physics, Calculus, and Mechanics of Material. The book is also a handy reference for researchers and professional engineers.

Leseprobe

Contents

Cover

Title Page

Copyright

About the Author

Preface

Chapter 1: Introduction to Structural Vibrations

1.1 Terminology

1.2 Types of Vibration

1.3 Objectives of Vibration Analyses

1.4 Global and Local Vibrations

1.5 Theoretical Approaches to Structural Vibrations

References

Chapter 2: Analytical Solutions for Uniform Continuous Systems

2.1 Methods for Obtaining Equations of Motion of a Vibrating System

2.2 Vibration of a Stretched String

2.3 Longitudinal Vibration of a Continuous Rod

2.4 Torsional Vibration of a Continuous Shaft

2.5 Flexural Vibration of a Continuous Euler–Bernoulli Beam

2.6 Vibration of Axial-Loaded Uniform Euler–Bernoulli Beam

2.7 Vibration of an Euler–Bernoulli Beam on the Elastic Foundation

2.8 Vibration of an Axial-Loaded Euler Beam on the Elastic Foundation

2.9 Flexural Vibration of a Continuous Timoshenko Beam

2.10 Vibrations of a Shear Beam and a Rotary Beam

2.11 Vibration of an Axial-Loaded Timoshenko Beam

2.12 Vibration of a Timoshenko Beam on the Elastic Foundation

2.13 Vibration of an Axial-Loaded Timoshenko Beam on the Elastic Foundation

2.14 Vibration of Membranes

2.15 Vibration of Flat Plates

References

Chapter 3: Analytical Solutions for Non-Uniform Continuous Systems: Tapered Beams

3.1 Longitudinal Vibration of a Conical Rod

3.2 Torsional Vibration of a Conical Shaft

3.3 Displacement Function for Free Bending Vibration of a Tapered Beam

3.4 Bending Vibration of a Single-Tapered Beam

3.5 Bending Vibration of a Double-Tapered Beam

3.6 Bending Vibration of a Nonlinearly Tapered Beam

References

Chapter 4: Transfer Matrix Methods for Discrete and Continuous Systems

4.1 Torsional Vibrations of Multi-Degrees-of-Freedom Systems

4.2 Lumped-Mass Model Transfer Matrix Method for Flexural Vibrations

4.3 Continuous-Mass Model Transfer Matrix Method for Flexural Vibrations

4.4 Flexural Vibrations of Beams with In-Span Rigid (Pinned) Supports

References

Chapter 5: Eigenproblem and Jacobi Method

5.1 Eigenproblem

5.2 Natural Frequencies, Natural Mode Shapes and Unit-Amplitude Mode Shapes

5.3 Determination of Normal Mode Shapes

5.4 Solution of Standard Eigenproblem with Standard Jacobi Method

5.5 Solution of Generalized Eigenproblem with Generalized Jacobi Method

5.6 Solution of Semi-Definite System with Generalized Jacobi Method

5.7 Solution of Damped Eigenproblem

References

Chapter 6: Vibration Analysis by Finite Element Method

6.1 Equation of Motion and Property Matrices

6.2 Longitudinal (Axial) Vibration of a Rod

6.4 Flexural Vibration of an Euler–Bernoulli Beam

6.5 Shape Functions for a Three-Dimensional Timoshenko Beam Element

6.6 Property Matrices of a Three-Dimensional Timoshenko Beam Element

6.7 Transformation Matrix for a Two-Dimensional Beam Element

6.8 Transformations of Element Stiffness Matrix and Mass Matrix

6.9 Transformation Matrix for a Three-Dimensional Beam Element

6.10 Property Matrices of a Beam Element with Concentrated Elements

6.11 Property Matrices of Rigid–Pinned and Pinned–Rigid Beam Elements

6.12 Geometric Stiffness Matrix of a Beam Element Due to Axial Load

6.13 Stiffness Matrix of a Beam Element Due to Elastic Foundation

References

Chapter 7: Analytical Methods and Finite Element Method for Free Vibration Analyses of Circularly Curved Beams

7.1 Analytical Solution for Out-of-Plane Vibration of a Curved Euler Beam

7.2 Analytical Solution for Out-of-Plane Vibration of a Curved Timoshenko Beam

7.3 Analytical Solution for In-Plane Vibration of a Curved Euler Beam

7.4 Analytical Solution for In-Plane Vibration of a Curved Timoshenko Beam

7.5 Out-of-Plane Vibration of a Curved Beam by Finite Element Method with Curved Beam Elements

7.6 In-Plane Vibration of a Curved Beam by Finite Element Method with Curved Beam Elements

7.7 Finite Element Method with Straight Beam Elements for Out-of-Plane Vibration of a Curved Beam

7.8 Finite Element Method with Straight Beam Elements for In-Plane Vibration of a Curved Beam

References

Chapter 8: Solution for the Equations of Motion

8.1 Free Vibration Response of an SDOF System

8.2 Response of an Undamped SDOF System Due to Arbitrary Loading

8.3 Response of a Damped SDOF System Due to Arbitrary Loading

8.4 Numerical Method for the Duhamel Integral

8.5 Exact Solution for the Duhamel Integral

8.6 Exact Solution for a Damped SDOF System Using the Classical Method

8.7 Exact Solution for an Undamped SDOF System Using the Classical Method

8.8 Approximate Solution for an SDOF Damped System by the Central Difference Method

8.9 Solution for the Equations of Motion of an MDOF System

8.10 Determination of Forced Vibration Response Amplitudes

8.11 Numerical Examples for Forced Vibration Response Amplitudes

References

Appendices

A.1 List of Integrals

A.2 Theory of Modified Half-Interval (or Bisection) Method

A.3 Determinations of Influence Coefficients

A.4 Exact Solution of a Cubic Equation

A.5 Solution of a Cubic Equation Associated with Its Complex Roots

A.6 Coefficients of Matrix [H] Defined by Equation (7.387)

A.7 Coefficients of Matrix [H] Defined by Equation (7.439)

A.8 Exact Solution for a Simply Supported Euler Arch

References

Index

This edition first published 2013

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Library of Congress Cataloging-in-Publication Data

Wu, Jong-Shyong.

Analytical and numerical methods for vibration analyses / Jong-Shyong Wu.

pages cm

Includes bibliographical references and index.

ISBN 978-1-118-63215-4 (hardback)

1. Vibration–Mathematical models. 2. Structural analysis(Engineering)–Mathematical models. I. Title.

TA355.W82 2013

620.301′51–dc23

2013008893

About the Author

Professor Jong-Shyong Wu, obtained his BS and MS degrees from the Department of Mechanical Engineering, National Cheng-Kung University (NCKU), Taiwan, in 1966 and 1969, respectively. After working in a factory for a year and a half, he was recommended to the Department of Naval Architecture and Marine Engineering (DNAME) of NCKU as an instructor, and he has stayed there until now, more than 40 years. During the period in NCKU, he obtained the financial support of the National Science Council (NSC), Taiwan, in 1975, and went abroad to study in the Department of Shipbuilding and Marine Technology, University of Strathclyde, UK, obtaining his Ph.D. degree in 1978. Because his thesis was related to ship hull vibrations in regular waves and in confused seas, he began to give courses on the dynamics of structures and the theory of matrix structural analysis to students in several graduate schools of the Engineering College, NCKU, after he returned to Taiwan in 1978. He has been the Chairman of DNAME of NCKU for 6 years and the head of the Computer and Network Center of NCKU for 6.5 years. He was promoted to professor in 1981 and awarded a distinguished professorship of NCKU in 2004. He has published more than 50 SCI papers in international journals, and obtained the prizes for Outstanding Research (three times) from NSC, the prize of Outstanding Engineering Paper from the Chinese Society of Engineers, Taiwan, and the Medal of Engineering from the Chinese Society of Naval Architects and Marine Engineers, Taiwan. He has been the supervisor of more than 70 MS students and seven Ph.D. students, and a reviewer for more than 15 international journals. This book is the condensation of his multi-year lecture notes and some new approaches that have appeared in his journal papers. It is suitable for undergraduate students, graduate students, engineers or researchers, dependent on the contents of the chapters concerned.

Preface

The author has given courses on the dynamics of structures and the theory of matrix structural analysis to students in several graduate schools (such as Mechanical, Civil, Aeronautical, Marine engineering, etc.) of the Engineering College, National Cheng-Kung University (NCKU), Taiwan, for more than 30 years. In addition to some reference books, lecture notes are the main teaching materials. Since the contents of lecture notes were usually changed annually, some students have asked me to write a book to cover the material in all my lecture notes over the past years. That is the main reason why I have written this book. In addition, it was also important to write a book to introduce the theories and methods presented in some of the author's publications appearing in international journals, despite the much trouble regarding the third-party permissions. The title of this book is Analytical and Numerical Methods for Vibration Analyses. It is obvious that the computer is one of the main tools for the solution of vibration problems, using either the analytical or the numerical methods introduced in this book. However, it is hoped that this book can also provide some useful information for readers who are not so familiar with computer languages or programming.

One of the predominant features of this book is that most of the introduced theories and associated mathematical expressions are confirmed by numerical examples. Most of the numerical results are obtained from two or three different methods, and good agreement between the numerical results of different methods is achieved. For example, in Chapter 4, the lowest five natural frequencies and corresponding mode shapes of a uniform or non-uniform beam carrying an arbitrary number of concentrated elements, including lumped masses (with eccentricities and rotary inertias), linear springs, rotational springs and spring–mass systems, are determined by three methods: the lumped-mass model transfer matrix method (LTMM); the continuous-mass model transfer matrix method (CTMM); and the conventional finite element method (FEM). In Chapter 7, for either out-of-plane or in-plane vibrations, the lowest five natural frequencies and associated mode shapes of a circularly curved beam carrying an arbitrary number of concentrated elements are also determined by three methods: the analytical (exact) method; the FEM with curved beam elements, FEM(curved); and the conventional FEM with straight beam elements, FEM(straight). In the other chapters, most of the numerical results are obtained from both the classical analytical (exact) method and the conventional FEM.

Since longitudinal and torsional vibration analyses are also important in the design of the propulsive shafting systems of ships, some attention is paid to the introduction of the axial vibrations of uniform and conical rods, and the torsional vibrations of uniform and conical shafts by using analytical methods, the TMM and the FEM. In addition, in existing books, the shape functions associated with the 12 degrees of freedom of the three-dimensional Timoshenko beam element are incomplete or neglected. Thus, in Chapter 6, much effort is devoted to their derivation, and then a consistent approach is used to determine the element stiffness matrix and consistent mass matrix by using these shape functions. Furthermore, in the existing literature, the forced vibration response “amplitudes” of a single-degree-of-freedom (SDOF) or a multi-degree-of-freedom (MDOF) system are determined from the “steady-state” responses, and the free vibrating effects appearing in the intermediate steps of forced vibrations are neglected. Since the amplitudes of the last classical “steady-state” responses are much less than the corresponding ones of “total” responses near resonance, in Chapter 8 an efficient technique for determining the amplitudes of “total” forced vibration responses of SDOF and MDOF systems is introduced, in addition to the theory for obtaining the amplitudes of classical “steady-state” responses. Numerical results reveal that the CPU time required by the introduced approach is less than 1% of that required by the conventional FEM. It is noted that a few statements or equations are repeated in this book for the convenience of readers.

At the moment of drinking water, one should think about where that water has come from. This is thus a suitable place for me to say “thanks” to some important people. I came from a farming family, where each member worked very hard. So, first of all, I must thank my father, mother, brothers and sister. It is because of their guidance and care since my childhood that I can overcome most difficulties. I am also greatly indebted to my wife (C.L. Chen) for her long-term support without complaint. Next, I would like to thank Mr. S.Y. Huang, president of Pin-Ho Iron Works, Inc., in whose factory I gained much practical experience. I am also grateful to Professor K.Y. Li, one of my teachers in the Department of Mechanical Engineering, NCKU; at his recommendation I became a teacher and begin my teaching career in the university. Definitely, he is one of my benefactors and will be remembered forever. My thanks also go to Professor C. Kuo in the Department of Shipbuilding and Marine Technology, University of Strathclyde, UK. My first paper in an international journal, “On wave-excited ship vibrations in regular waves and in confused seas”, was finished under his supervision. Of course, the financial support of the National Science Council (NSC), Taiwan, leading to my Ph.D. is highly appreciated. I would also like to thank Professor J.R. Maa, one of the former presidents of NCKU – he is a respected scholar and officer, and I learnt a lot from him about many things. Particular thanks go to the following for their help and encouragement: Professors C.I. Weng, C.K. Chen, S.J. Hsieh, T.S. Wang, T.S. Su, P.A. Luh, M.L. Lee, M.J. Huang, H.J. Shaw, C.H. Huang, R.M. Chao, G.P. Too, J.M. Yang, D.S. Hsu, J.Q. Tarn, S.Y. Tsai, H.H. Hwung, Y.L. Chou, T.S. Chen, C.Y. Yang, M.H. Wu, C.C. Liang, T.L. Teng, C. Kao, C. Hsu, W.H. Wang, R.J. Shyu, Y.B. Yang, Y.J. Lee, Y.H. Chen, C.F. Hung, J.H. Kuang and H.Y. Lin, and Messrs. S.P. Lin, T.S. Chen, R.G. Hung, F.S. Wu, J.H. Wu, C. Wu and C.Y. Chou. Finally, from August 1990 to February 1997, I worked in the Computer and Network Center of NCKU, and the people there who helped me very much are highly appreciated: Professors Y.N. Sun, S.R. Tsai, J.H. Chou, J.F. Wang, C.K. Shieh and S.J. Wang, and teachers R.L. Wu, J.W. Yan and N.H. Chiang.

Some materials in this book come from projects supported by the NSC, and this book was finished in the pleasant environment provided by NCKU, both of which are much appreciated. Of course, I must also say “thanks” to many of my other colleagues, friends and students, although they are not mentioned above because of space limitations. Finally, I would like to give my best regards to the anonymous reviewers for their valuable comments on and kind advice about my publications.

Jong-Shyong Wu

January 2013

1

Introduction to Structural Vibrations

For the convenience of subsequent descriptions, some terminology is introduced first. Then, the lowest 20 natural frequencies and associated mode shapes of a three-dimensional free–free (F-F) uniform beam obtained from the conventional finite element method (FEM) are used to show the types of vibrations: rigid-body motions and elastic vibrations. In general, the so-called vibrations refer to elastic vibrations, including longitudinal (or axial), torsional and bending vibrations. Furthermore, bending vibrations are also called flexural, transverse or lateral vibrations. For practice, the natural frequency chart for a ship under light and heavy loading conditions is used to illustrate the purpose of general free vibration analysis. The time histories and frequency-response (amplitude) curves of three points on a uniform beam with pinned–pinned supporting conditions and subjected to a vertical harmonic exciting force are used to indicate the main objective of forced vibration analysis. For a complicated system, either global vibration or local vibration or both may be important for the solution of the vibration problem concerned, and thus some relevant information is also introduced. Finally, the contents of this book are briefly outlined.

1.1 Terminology

(a) Degrees of Freedom

The minimum number of independent coordinates required to define the configuration of a dynamic system at any instant of time is called the degrees of freedom (DOFs) of the system. Since the instantaneous position of an unconstrained particle in space can be determined by three independent coordinates (, and ), it has three DOFs in space. Similarly, the instantaneous position of an unconstrained rigid body in space can be determined by six independent coordinates (, , , , and ), and it has six DOFs in space. Here the symbols , and denote the translational (or linear) displacements of the center of gravity of the rigid body with respect to the reference coordinate system , while , and denote the rotational (or angular) displacements of the rigid body about the x, y and z axes, respectively.

(b) Free Vibration

The vibration of a system occurring in the absence of forced vibration is called free vibration. It is a phenomenon of repeated oscillations induced by the initial displacement(s) and/or initial velocity (or velocities) of the system.

(c) Natural Frequency

The number of repeated oscillations per unit time is called the natural frequency for a free vibration system. The total number of natural frequencies for a free vibration system is equal to the total number of DOFs of the system, among which the lowest one is called the fundamental frequency. The unit of natural frequency obtained from the characteristic (or eigenproblem) equation of a vibrating system is “radians per second” (rad/s). The frequency with unit “rad/s” is called circular frequency (or angular frequency) and denoted by . For convenience, the cyclic frequency with unit “cycles per second” (cps or Hz) is frequently used and denoted by f. The relationship between circular frequency and cyclic frequency is given by or . Another commonly used unit of natural frequency is “revolutions per minute” (RPM) and denoted by N. It is evident that .

(d) Forced Vibration

The oscillation of a system induced by external dynamic loads is called forced vibration. Because an external load is a vector possessing magnitude, direction and position, any load with time-dependent magnitude, direction and/or position is a dynamic load. It is obvious that the oscillation of a system subjected to a dynamic load is time-dependent.

(e) Resonance

The dynamic response amplitude of a forced vibration system is dependent on many factors, among which the exciting frequency of the external load is the most important one. The (global or local) maximum response amplitude of a vibrating system will appear when approaches any of the natural frequencies of the system (particularly when ). This dynamic condition is called resonance of the vibrating system.

(f) Damping and Critical Damping Coefficient

In the natural environment, the vibration amplitude of a free vibration system will decay with time and finally stop. This is because the kinetic energy of the vibrating system is consumed by the surrounding friction or resistance. The mechanism that can consume the energy of a vibration system is called damping. In general, the damping of most vibrating systems is small and its influence on the natural frequencies or forced vibration responses is negligible. However, a slight damping can significantly reduce the maximum response amplitude of a vibrating system in the resonant condition. In other words, the damping is important for a vibrating system near resonance. A single-degree-of-freedom (SDOF) system can perform free vibration only if its damping coefficient is less than a critical value , with k and m denoting its stiffness and mass, respectively. The value of is called the critical damping coefficient of the vibrating system. If c denotes the actual damping coefficient of a SDOF system, then the system is an under-damped, critically damped or over-damped system dependent on whether , or . It is noted that only an under-damped system can perform free vibrations. For a system with and subjected to an impact load, it will depart from its static equilibrium position (SEP) and then gradually return to its original SEP. In such a case, no free vibration of the system will occur.

(g) Periodic Motion and Harmonic Motion

If the same motion of a system repeats with the same time interval , this kind of motion is called periodic motion and the time interval is called the period of the motion. Based on the definition of the cyclic frequency , one has or . A periodic motion may be simple or complex. A simple periodic motion with displacement taking the form , or is called harmonic motion, and and are the amplitude and natural frequency of the vibrating system, respectively.