Structural Dynamics

Madhujit Mukhopadhyay

Structural DynamicsVibrations and Systems

Madhujit MukhopadhyayIndian Institute of Technology KharagpurKharagpur, India

ISBN 978-3-030-69673-3 ISBN 978-3-030-69674-0 (eBook)https://doi.org/10.1007/978-3-030-69674-0

Jointly published with ANE Books Pvt. Ltd.In addition to this printed edition, there is a local printed edition of this work available via Ane Books inSouth Asia (India, Pakistan, Sri Lanka, Bangladesh, Nepal and Bhutan) and Africa (all countries in theAfrican subcontinent).ISBN of the Co-Publisher’s edition: 9788180520907.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer NatureSwitzerland AG 2021This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whetherthe whole or part of the material is concerned, specifically the rights of translation, reprinting, reuseof illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, andtransmission or information storage and retrieval, electronic adaptation, computer software, or by similaror dissimilar methodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.The publishers, the authors, and the editors are safe to assume that the advice and information in this bookare believed to be true and accurate at the date of publication. Neither the publishers nor the authors orthe editors give a warranty, express or implied, with respect to the material contained herein or for anyerrors or omissions that may have been made. The publishers remain neutral with regard to jurisdictionalclaims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Switzerland AGThe registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

The book is dedicated

to the memory of

The Late Professor Bidhu Ranjan Sen

Preface

During last three decades the subject “Vibrations and Dynamics of Structural andMechanical Systems” has undergone significant development. There has been an urgeand urgency of designing present day modern and complex structures and systemsin their proper perspective, a task which could not be conceived even three decadesback, This has been possible due to the advent of the electronic digital computer.With the exception of the self weight of the structures, strictly speaking, no otherload can be treated as static load. In order to truly analyse these systems, a knowledgeof treatment the dynamic load is of paramount importance.

The existing book on the subject mainly deal with one of the following aspects:

1. Vibrations of Mechanical Systems.2. Structural Dynamics.

Though basic principles and solution techniques are same for both the abovetopics, they primarily differ in their application areas. The main objective of the bookis to present these aspects in a unified and integrated manner. The chapters have beenarranged in such a way that starting from the basics, the reader is acquainted withmore advanced topics in a gradual manner.

The material of the book has been developed while teaching the students of IndianInstitute of Technology, Kharagpur. A similar book was written by the present authorwhich is no longer available in themarket and also it iswithdrawn.As such a necessityhas arisen for the publication of the present book.

The author is indebted to a number of persons, withoutwhose help, the preparationof the manuscript would not have been possible. To name the most important personsamongst them are—Das, Prabir Sinha, Parimal Kumar Roy, ChinmoyMukherjee andRatna, the wife of the author.

Lastly, the author expresses his sincere thanks toMr. Sunil Saxena, Jai Raj Kapoorand Sudipta Ghosh—all of Ane Books, for their help and the excellent cooperationreceived.

Kharagpur, India Madhujit Mukhopadhyay

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Brief History of Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Comparison Between Static and Dynamic Analysis . . . . . . . . . . 41.4 D’alembert’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Some Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5.1 Vibration and Oscillation . . . . . . . . . . . . . . . . . . . . . . . . 61.5.2 Free Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5.3 Forced Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5.4 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5.5 Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6 Dynamic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.7 Finite Element Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.8 Response of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.9 Types of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.10 Linear and Nonlinear Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Free Vibration of Single Degree of Freedom System . . . . . . . . . . . . . . 152.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Equation of Motion of Single Degree of Freedom (Sdf)

System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Free Undamped Vibration of the Sdf System . . . . . . . . . . . . . . . . 182.4 Free Damped Vibration of Sdf System . . . . . . . . . . . . . . . . . . . . . 282.5 Free Vibration with Coulomb Damping . . . . . . . . . . . . . . . . . . . . . 342.6 Energy Method and Free Torsional Vibration . . . . . . . . . . . . . . . . 37

2.6.1 Torsional Vibration of the SDF System . . . . . . . . . . . . 382.6.2 Rayleigh’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.7 Logarithmic Decrement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

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3 Forced Vibration of Single Degree of Freedom System . . . . . . . . . . . . 593.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2 Response of Damped Systems to Harmonic Loading . . . . . . . . . 593.3 Rotating Unbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.4 Reciprocating Unbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.5 Whirling of Rotating Shafts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.6 Vibration Isolation and Transmissibility . . . . . . . . . . . . . . . . . . . . 73

3.6.1 Transmissibility Due to Support Motions . . . . . . . . . . . 753.7 Energy Dissipation by Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.8 Equivalent Viscous Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.9 Self-excited Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.10 Vibration Measuring Seismic Instruments . . . . . . . . . . . . . . . . . . . 86

3.10.1 Vibrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.10.2 Accelerometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.10.3 Discussion About the Instruments . . . . . . . . . . . . . . . . . 89

3.11 Response of Structures Due to Transient Vibration . . . . . . . . . . . 903.11.1 Response of SDF System to an Ideal Step Input . . . . . 903.11.2 Response of SDF System to Gradually Applied

Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.12 Response to SDF Systems to a General Type of Forcing

Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.13 Dynamic Load Factor and Response Spectrum . . . . . . . . . . . . . . 973.14 Response Due to Periodic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.14.1 Real Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983.14.2 Response of SDF System to Periodic Forces

Represented by Real Fourier Series . . . . . . . . . . . . . . . 1013.14.3 Complex Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.14.4 Response of SDF System to Periodic Forces

Represented by Complex Fourier Series . . . . . . . . . . . . 1043.15 Response Due to Non-periodic Excitation . . . . . . . . . . . . . . . . . . . 1063.16 Relationship Between Complex Frequency Response

Function and Unit Impulse Response Function . . . . . . . . . . . . . . 1093.17 Support Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.17.1 Displacement Approach . . . . . . . . . . . . . . . . . . . . . . . . . 1113.17.2 Acceleration Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3.18 Response of SDF Systems Related to Earthquakes . . . . . . . . . . . 1133.19 Techniques for Analysing Earthquake Response . . . . . . . . . . . . . 115References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4 Numerical Methods in Structural Dynamics: Applied to SDFSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.2 Direct Integration Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.2.1 Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . 1304.2.2 Linear Acceleration Method . . . . . . . . . . . . . . . . . . . . . . 135

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4.2.3 Runge–Kutta Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 1384.2.4 Newmark’s β-Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

4.3 Numerical Evaluation of Duhamel’s Integral . . . . . . . . . . . . . . . . 1424.3.1 Numerical Evaluation of Damped System

by Duhamel’s Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 1454.4 Numerical Computation in Frequency Domain . . . . . . . . . . . . . . 148

4.4.1 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 1484.4.2 Fast Fourier Transform (FFT) . . . . . . . . . . . . . . . . . . . . 149

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

5 Vibration of Two Degrees of Freedom System . . . . . . . . . . . . . . . . . . . . 1555.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.2 Free Vibration of Undamped Two Degrees of Freedom

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.3 Torsional Vibration of Two Degrees of Freedom System . . . . . . 1595.4 Forced Vibration of Two Degrees of Freedom Undamped

System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1615.5 Vibration Absorber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1625.6 Free Vibration of Two Degrees of Freedom System

with Viscous Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1675.7 Coordinate Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1685.8 Free Vibration of Damped Two Degrees of Freedom

System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

6 Free Vibration of Multiple Degrees of Freedom System . . . . . . . . . . . 1836.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1836.2 Equations of Motion of MDF Systems . . . . . . . . . . . . . . . . . . . . . 183

6.2.1 Mass Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1856.2.2 The Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1866.2.3 The Damping Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1866.2.4 Loading Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

6.3 Free Undamped Vibration Analysis of MDF Systems . . . . . . . . . 1866.4 Orthogonality Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1886.5 Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1906.6 Determination of Absolute Displacement of Free

Vibration of MDF Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1916.6.1 Normalisation of Modes . . . . . . . . . . . . . . . . . . . . . . . . . 196

6.7 Eigenvalue Solution Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 1966.8 Dunkerley’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1986.9 Holzer Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2006.10 Transfer Matrix Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

6.10.1 Transfer Matrices as a Means of Elimination . . . . . . . 2076.11 Myklestad Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2116.12 Stodola’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2176.13 Matrix Deflation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

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6.14 Rayleigh’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2296.15 Rayleigh–Ritz Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2316.16 Subspace Iteration Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2336.17 Simultaneous Iteration Method and Algorithm . . . . . . . . . . . . . . . 2346.18 Geared Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2356.19 Branched Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2366.20 Reduction Methods for Dynamic Analysis . . . . . . . . . . . . . . . . . . 238

6.20.1 Static Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2396.20.2 Guyan Reduction Method of Dynamic Analysis . . . . . 240

6.21 Component Mode Synthesis Method . . . . . . . . . . . . . . . . . . . . . . . 2446.21.1 Fixed Interface Method . . . . . . . . . . . . . . . . . . . . . . . . . . 2446.21.2 Free Interface Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

6.22 Lagrange’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

7 Forced Vibration Analysis of Multiple Degrees of FreedomSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2617.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2617.2 Mode Superposition Method for the Determination

of Response of Undamped MDF System . . . . . . . . . . . . . . . . . . . . 2617.3 Mode-Acceleration Method for the Determination

of Response of MDF System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2657.4 Response of MDF Systems Under the Action of Transient

Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2677.5 Damping in MDF Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

7.5.1 Conditions for Damping Uncoupling . . . . . . . . . . . . . . 2767.5.2 Extended Rayleigh Damping . . . . . . . . . . . . . . . . . . . . . 280

7.6 Response of MDF Systems to Support Motion . . . . . . . . . . . . . . . 2827.7 Earthquake Spectrum Analysis of Structures Having

MDF System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2867.8 Use of Response Spectra for Designing MDF Systems . . . . . . . . 2887.9 Direct Integration for Determining Response of MDF

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2927.10 Complex Matrix Inversion Method for Forced Vibration

Analysis of MDF Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2967.11 Frequency Domain Analysis of MDF Systems by Modal

Superposition for Harmonic Loads . . . . . . . . . . . . . . . . . . . . . . . . . 2977.12 Frequency Domain Analysis of Direct Frequency

Response Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

8 Free Vibration Analysis of Continuous Systems . . . . . . . . . . . . . . . . . . 3078.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3078.2 Vibration of Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

8.2.1 Wave Propagation Solution . . . . . . . . . . . . . . . . . . . . . . 3118.3 Free Longitudinal Vibration of a Bar . . . . . . . . . . . . . . . . . . . . . . . 314

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8.3.1 Free Longitudinal Vibration of a Bar Clampedat X = 0 and Free at X = L . . . . . . . . . . . . . . . . . . . . . . . 317

8.4 Free Torsional Vibration of the Shaft . . . . . . . . . . . . . . . . . . . . . . . 3188.5 Free Flexural Vibration of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . 3228.6 Free Flexural Vibration of the Simply Supported Beam . . . . . . . 3248.7 Free Flexural Vibration of Beams with Other End

Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3268.7.1 Uniform Beam Having Both Ends Free . . . . . . . . . . . . 326

8.8 Free Flexural Vibration of Beams with General EndConditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

8.9 Orthogonality Properties of Normal Modes . . . . . . . . . . . . . . . . . 3368.10 Effect of Rotary Inertia on the Free Flexural Vibration

of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3398.11 Free Vibration of the Shear Beam . . . . . . . . . . . . . . . . . . . . . . . . . . 3438.12 Effect of Axial Force on the Free Flexural Vibration

of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3458.13 Free Vibration of Beams Including Shear Deformation

and Rotary Inertia Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3478.14 Collocation Method for Obtaining Normal Modes

of Vibration of a Continuous System . . . . . . . . . . . . . . . . . . . . . . . 3508.15 Rayleigh’s Quotient for Fundamental Frequency . . . . . . . . . . . . . 3548.16 Rayleigh–Ritz Method for Determining Natural

Frequencies of Continuous Systems . . . . . . . . . . . . . . . . . . . . . . . . 3568.17 Vibration of Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3598.18 Transverse Vibration of Rectangular Thin Plates . . . . . . . . . . . . . 363References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

9 Forced Vibration of Continuous Systems . . . . . . . . . . . . . . . . . . . . . . . . 3719.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3719.2 Forced Axial Vibration of Bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3719.3 Forced Vibration of the Shear Beam Under Ground

Motion Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3759.4 Forced Vibration of Flexural Member . . . . . . . . . . . . . . . . . . . . . . 3789.5 Forced Transverse Vibration of Uniform Damped Beam . . . . . . 3829.6 Forced Vibration of Flexural Member Subjected

to Ground Motion Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3849.7 Response of Beams Due to Moving Loads . . . . . . . . . . . . . . . . . . 386

9.7.1 Response of the Beam When the Massof the Vehicle is Large . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

9.7.2 Response of the Beam when the Massof the Vehicle is Small . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

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10 Dynamic Direct Stiffness Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39510.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39510.2 Continuous Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39510.3 Methods Analogous to Classical Methods in Statical

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39810.4 Dynamic Stiffness Matrix in Bending . . . . . . . . . . . . . . . . . . . . . . 40110.5 Dynamic Stiffness Matrix for Flexure and Rigid Axial

Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40810.6 Dynamic Stiffness Matrix of a Bar Undergoing Axial

Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41110.7 Dynamic Stiffness Matrix of a Bar Subjected to Axial

and Bending Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41310.7.1 Combined Uncoupled Axial and Bending

Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41310.7.2 Combined Coupled Axial and Bending Stiffness . . . . 414

10.8 Beam Segments with Distributed Mass Having ShearDeformation and Rotary Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

11 Vibration of Ship and Aircraft as a Beam . . . . . . . . . . . . . . . . . . . . . . . . 42511.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42511.2 Shift in Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42511.3 Added Mass of a Ship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42611.4 The Flexibility Matrix Method for Determining Natural

Frequencies of a Free-Free Beam in Vertical Vibration . . . . . . . . 42811.5 The Flexibility Matrix Method for the Analysis

of Coupled Horizontal and Torsional Vibration . . . . . . . . . . . . . . 43311.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

12 Finite Element Method in Vibration Analysis . . . . . . . . . . . . . . . . . . . . 44312.1 Introduction to the Finite Element Method . . . . . . . . . . . . . . . . . . 44312.2 Torsional Vibration of Shafts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44312.3 Axial Vibration of Rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44812.4 Flexural Vibration of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45112.5 Vibration of Timoshenko Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . 45512.6 Inplane Vibration of Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

12.6.1 Linear Triangular Element . . . . . . . . . . . . . . . . . . . . . . . 45912.6.2 Linear Rectangular Element . . . . . . . . . . . . . . . . . . . . . . 463

12.7 Flexural Vibration of Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46612.8 Flexural Vibrations of Plates Using Isoparametric

Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46912.8.1 Reduced Integration Technique . . . . . . . . . . . . . . . . . . . 47312.8.2 Consistent Mass Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 474

12.9 Periodic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47612.9.1 Different Types of Periodic Structures . . . . . . . . . . . . . 477

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12.9.2 Free Harmonic Wave Motion Througha Mono-coupled Periodic Beam . . . . . . . . . . . . . . . . . . . 478

12.9.3 Finite Element Analysis of Periodic Structures . . . . . . 483References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

13 Finite Difference Method for the Vibration Analysis of Beamsand Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49113.1 Introduction to the Finite Difference Method . . . . . . . . . . . . . . . . 49113.2 Central Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49113.3 Free Vibration of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49313.4 Free Vibration of Rectangular Plates . . . . . . . . . . . . . . . . . . . . . . . 49613.5 Semi-analytic Finite Difference Method for Free

Vibration Analysis of Rectangular Plates . . . . . . . . . . . . . . . . . . . 49713.6 Semi-analytic Finite Difference Method for Forced

Vibration Analysis of Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50113.7 Spline Finite Strip Method of Analysis of Plate Vibration . . . . . 503

13.7.1 The Spline Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50413.7.2 Displacement Functions . . . . . . . . . . . . . . . . . . . . . . . . . 50613.7.3 Strain–Displacement Relationship . . . . . . . . . . . . . . . . 50713.7.4 Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50713.7.5 Consistent Mass Matrix of the Plate Strip . . . . . . . . . . 508

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

14 Nonlinear Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51514.2 Perturbation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

14.2.1 Step-By-Step Integration . . . . . . . . . . . . . . . . . . . . . . . . 520References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

15 Random Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52715.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52715.2 Random Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52715.3 Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528

15.3.1 Second-Order Probability Distribution . . . . . . . . . . . . . 53015.4 Ensemble Averages, Mean and Autocorrelation . . . . . . . . . . . . . . 53215.5 Stationary Process, Ergodic Process and Temporal

Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53515.5.1 Stationary Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53515.5.2 Temporal Statistics and Ergodic Hypothesis . . . . . . . . 536

15.6 Power Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53915.7 Relationship Between Autocorrelation Function

and Power Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54015.8 Random Response of SDF Systems . . . . . . . . . . . . . . . . . . . . . . . . 543

15.8.1 Time Domain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 54415.8.2 Frequency Domain Analysis . . . . . . . . . . . . . . . . . . . . . 544

15.9 Random Response of MDF Systems . . . . . . . . . . . . . . . . . . . . . . . 546

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15.9.1 Complex Matrix Inversion Method . . . . . . . . . . . . . . . . 54715.9.2 Normal Mode Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

15.10 Response of Flexural Beams Under Random Loading . . . . . . . . 55215.11 Finite Element Random Response of Plates . . . . . . . . . . . . . . . . . 553

15.11.1 Cross-Spectral Density Matrix for GeneralisedForces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555

15.11.2 Response Spectra for Displacementsand Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

16 Computer Programs in Vibration Analysis . . . . . . . . . . . . . . . . . . . . . . 56316.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56316.2 Computer Program for Forced Vibration Analysis . . . . . . . . . . . . 56316.3 Computer Program for Random Vibration Analysis . . . . . . . . . . 56416.4 Computer Program for Free Vibration Analysis of Framed

Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56616.4.1 A Plane Frame Problem . . . . . . . . . . . . . . . . . . . . . . . . . 582

16.5 Computer Program for the Free Vibration Analysisof Ships by Flexibility Matrix Method . . . . . . . . . . . . . . . . . . . . . . 584

16.6 Computer Program for Finite Element Free VibrationAnalysis of Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59316.6.1 A Plate Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

Appendix A: The Stiffness Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

Appendix B: Table of Spring Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619

About the Author

Prof. Madhujit Mukhopadhyay is Ex-Dean (Faculty and Planning) and ProfessorEmeritus of Structural Engineering at Indian Institute of Technology, Kharagpur. Heobtained the B.E. degree from the University of Calcutta and Ph.D. and the D.Sc.degrees from IIT, Kharagpur. His field of research interest lies in the application ofvarious numerical methods for the analysis of plates and shells, bare of stiffened,isotropic or orthotropic. He has received several awards for his research. A widelytraveled person, Prof. Madhujit Mukhopadhyay published a large number of papersin international journals in aerospace, civil, mechanical and ocean engineering andis the author of four textbooks.

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