Structural Dynamics : Vibrations and Systems

This book introduces the theory of structural dynamics, with focus on civil engineering structures. It presents modern methods of analysis and techniques adaptable to computer programming clearly and easily. The book is ideal as a text for advanced undergraduates or graduate students taking a first course in structural dynamics. It is arranged in such a way that it can be used for a one- or two-semester course, or span the undergraduate and graduate levels. In addition, this book serves the practicing engineer as a primary reference. This book is organized by the type of structural modeling. The author simplifies the subject by presenting a single degree-of-freedom system in the first chapters and then moves to systems with many degrees-of-freedom in the following chapters. Many worked examples/problems are presented to explain the text, and a few computer programs are presented to help better understand the concepts. The book is useful to the research scholars and professional engineers, besides senior undergraduate and postgraduate students. Preface Contents About the Author 1 Introduction 1.1 Introduction 1.2 Brief History of Vibrations 1.3 Comparison Between Static and Dynamic Analysis 1.4 D’alembert’s Principle 1.5 Some Basic Definitions 1.5.1 Vibration and Oscillation 1.5.2 Free Vibration 1.5.3 Forced Vibration 1.5.4 Damping 1.5.5 Degrees of Freedom 1.6 Dynamic Loading 1.7 Finite Element Discretization 1.8 Response of the System 1.9 Types of Analysis 1.10 Linear and Nonlinear Vibration References 2 Free Vibration of Single Degree of Freedom System 2.1 Introduction 2.2 Equation of Motion of Single Degree of Freedom (Sdf) System 2.3 Free Undamped Vibration of the Sdf System 2.4 Free Damped Vibration of Sdf System 2.5 Free Vibration with Coulomb Damping 2.6 Energy Method and Free Torsional Vibration 2.6.1 Torsional Vibration of the SDF System 2.6.2 Rayleigh’s Method 2.7 Logarithmic Decrement References 3 Forced Vibration of Single Degree of Freedom System 3.1 Introduction 3.2 Response of Damped Systems to Harmonic Loading 3.3 Rotating Unbalance 3.4 Reciprocating Unbalance 3.5 Whirling of Rotating Shafts 3.6 Vibration Isolation and Transmissibility 3.6.1 Transmissibility Due to Support Motions 3.7 Energy Dissipation by Damping 3.8 Equivalent Viscous Damping 3.9 Self-excited Vibrations 3.10 Vibration Measuring Seismic Instruments 3.10.1 Vibrometer 3.10.2 Accelerometers 3.10.3 Discussion About the Instruments 3.11 Response of Structures Due to Transient Vibration 3.11.1 Response of SDF System to an Ideal Step Input 3.11.2 Response of SDF System to Gradually Applied Load 3.12 Response to SDF Systems to a General Type of Forcing Function 3.13 Dynamic Load Factor and Response Spectrum 3.14 Response Due to Periodic Forces 3.14.1 Real Fourier Series 3.14.2 Response of SDF System to Periodic Forces Represented by Real Fourier Series 3.14.3 Complex Fourier Series 3.14.4 Response of SDF System to Periodic Forces Represented by Complex Fourier Series 3.15 Response Due to Non-periodic Excitation 3.16 Relationship Between Complex Frequency Response Function and Unit Impulse Response Function 3.17 Support Motion 3.17.1 Displacement Approach 3.17.2 Acceleration Approach 3.18 Response of SDF Systems Related to Earthquakes 3.19 Techniques for Analysing Earthquake Response References 4 Numerical Methods in Structural Dynamics: Applied to SDF Systems 4.1 Introduction 4.2 Direct Integration Techniques 4.2.1 Finite Difference Method 4.2.2 Linear Acceleration Method 4.2.3 Runge–Kutta Method 4.2.4 Newmark’s β-Method 4.3 Numerical Evaluation of Duhamel’s Integral 4.3.1 Numerical Evaluation of Damped System by Duhamel’s Integral 4.4 Numerical Computation in Frequency Domain 4.4.1 Discrete Fourier Transform 4.4.2 Fast Fourier Transform (FFT) References 5 Vibration of Two Degrees of Freedom System 5.1 Introduction 5.2 Free Vibration of Undamped Two Degrees of Freedom Systems 5.3 Torsional Vibration of Two Degrees of Freedom System 5.4 Forced Vibration of Two Degrees of Freedom Undamped System 5.5 Vibration Absorber 5.6 Free Vibration of Two Degrees of Freedom System with Viscous Damping 5.7 Coordinate Coupling 5.8 Free Vibration of Damped Two Degrees of Freedom System References 6 Free Vibration of Multiple Degrees of Freedom System 6.1 Introduction 6.2 Equations of Motion of MDF Systems 6.2.1 Mass Matrix 6.2.2 The Stiffness Matrix 6.2.3 The Damping Matrix 6.2.4 Loading Matrix 6.3 Free Undamped Vibration Analysis of MDF Systems 6.4 Orthogonality Relationship 6.5 Eigenvalue Problem 6.6 Determination of Absolute Displacement of Free Vibration of MDF Systems 6.6.1 Normalisation of Modes 6.7 Eigenvalue Solution Techniques 6.8 Dunkerley’s Equation 6.9 Holzer Method 6.10 Transfer Matrix Method 6.10.1 Transfer Matrices as a Means of Elimination 6.11 Myklestad Method 6.12 Stodola’s Method 6.13 Matrix Deflation Procedure 6.14 Rayleigh’s Method 6.15 Rayleigh–Ritz Method 6.16 Subspace Iteration Method 6.17 Simultaneous Iteration Method and Algorithm 6.18 Geared Systems 6.19 Branched Systems 6.20 Reduction Methods for Dynamic Analysis 6.20.1 Static Condensation 6.20.2 Guyan Reduction Method of Dynamic Analysis 6.21 Component Mode Synthesis Method 6.21.1 Fixed Interface Method 6.21.2 Free Interface Method 6.22 Lagrange’s Equation References 7 Forced Vibration Analysis of Multiple Degrees of Freedom System 7.1 Introduction 7.2 Mode Superposition Method for the Determination of Response of Undamped MDF System 7.3 Mode-Acceleration Method for the Determination of Response of MDF System 7.4 Response of MDF Systems Under the Action of Transient Forces 7.5 Damping in MDF Systems 7.5.1 Conditions for Damping Uncoupling 7.5.2 Extended Rayleigh Damping 7.6 Response of MDF Systems to Support Motion 7.7 Earthquake Spectrum Analysis of Structures Having MDF System 7.8 Use of Response Spectra for Designing MDF Systems 7.9 Direct Integration for Determining Response of MDF Systems 7.10 Complex Matrix Inversion Method for Forced Vibration Analysis of MDF Systems 7.11 Frequency Domain Analysis of MDF Systems by Modal Superposition for Harmonic Loads 7.12 Frequency Domain Analysis of Direct Frequency Response Method References 8 Free Vibration Analysis of Continuous Systems 8.1 Introduction 8.2 Vibration of Strings 8.2.1 Wave Propagation Solution 8.3 Free Longitudinal Vibration of a Bar 8.3.1 Free Longitudinal Vibration of a Bar Clamped at X = 0 and Free at X = L 8.4 Free Torsional Vibration of the Shaft 8.5 Free Flexural Vibration of Beams 8.6 Free Flexural Vibration of the Simply Supported Beam 8.7 Free Flexural Vibration of Beams with Other End Conditions 8.7.1 Uniform Beam Having Both Ends Free 8.8 Free Flexural Vibration of Beams with General End Conditions 8.9 Orthogonality Properties of Normal Modes 8.10 Effect of Rotary Inertia on the Free Flexural Vibration of Beams 8.11 Free Vibration of the Shear Beam 8.12 Effect of Axial Force on the Free Flexural Vibration of Beams 8.13 Free Vibration of Beams Including Shear Deformation and Rotary Inertia Effects 8.14 Collocation Method for Obtaining Normal Modes of Vibration of a Continuous System 8.15 Rayleigh’s Quotient for Fundamental Frequency 8.16 Rayleigh–Ritz Method for Determining Natural Frequencies of Continuous Systems 8.17 Vibration of Membranes 8.18 Transverse Vibration of Rectangular Thin Plates References 9 Forced Vibration of Continuous Systems 9.1 Introduction 9.2 Forced Axial Vibration of Bars 9.3 Forced Vibration of the Shear Beam Under Ground Motion Excitation 9.4 Forced Vibration of Flexural Member 9.5 Forced Transverse Vibration of Uniform Damped Beam 9.6 Forced Vibration of Flexural Member Subjected to Ground Motion Excitation 9.7 Response of Beams Due to Moving Loads 9.7.1 Response of the Beam When the Mass of the Vehicle is Large 9.7.2 Response of the Beam when the Mass of the Vehicle is Small References 10 Dynamic Direct Stiffness Method 10.1 Introduction 10.2 Continuous Beam 10.3 Methods Analogous to Classical Methods in Statical Analysis 10.4 Dynamic Stiffness Matrix in Bending 10.5 Dynamic Stiffness Matrix for Flexure and Rigid Axial Displacements 10.6 Dynamic Stiffness Matrix of a Bar Undergoing Axial Deformation 10.7 Dynamic Stiffness Matrix of a Bar Subjected to Axial and Bending Deformations 10.7.1 Combined Uncoupled Axial and Bending Stiffness 10.7.2 Combined Coupled Axial and Bending Stiffness 10.8 Beam Segments with Distributed Mass Having Shear Deformation and Rotary Inertia References 11 Vibration of Ship and Aircraft as a Beam 11.1 Introduction 11.2 Shift in Stiffness Matrix 11.3 Added Mass of a Ship 11.4 The Flexibility Matrix Method for Determining Natural Frequencies of a Free-Free Beam in Vertical Vibration 11.5 The Flexibility Matrix Method for the Analysis of Coupled Horizontal and Torsional Vibration 11.6 Numerical Examples References 12 Finite Element Method in Vibration Analysis 12.1 Introduction to the Finite Element Method 12.2 Torsional Vibration of Shafts 12.3 Axial Vibration of Rods 12.4 Flexural Vibration of Beams 12.5 Vibration of Timoshenko Beams 12.6 Inplane Vibration of Plates 12.6.1 Linear Triangular Element 12.6.2 Linear Rectangular Element 12.7 Flexural Vibration of Plates 12.8 Flexural Vibrations of Plates Using Isoparametric Elements 12.8.1 Reduced Integration Technique 12.8.2 Consistent Mass Matrix 12.9 Periodic Structures 12.9.1 Different Types of Periodic Structures 12.9.2 Free Harmonic Wave Motion Through a Mono-coupled Periodic Beam 12.9.3 Finite Element Analysis of Periodic Structures References 13 Finite Difference Method for the Vibration Analysis of Beams and Plates 13.1 Introduction to the Finite Difference Method 13.2 Central Difference Method 13.3 Free Vibration of Beams 13.4 Free Vibration of Rectangular Plates 13.5 Semi-analytic Finite Difference Method for Free Vibration Analysis of Rectangular Plates 13.6 Semi-analytic Finite Difference Method for Forced Vibration Analysis of Plates 13.7 Spline Finite Strip Method of Analysis of Plate Vibration 13.7.1 The Spline Functions 13.7.2 Displacement Functions 13.7.3 Strain–Displacement Relationship 13.7.4 Stiffness Matrix 13.7.5 Consistent Mass Matrix of the Plate Strip References 14 Nonlinear Vibration 14.1 Introduction 14.2 Perturbation Method 14.2.1 Step-By-Step Integration References 15 Random Vibrations 15.1 Introduction 15.2 Random Process 15.3 Probability Distributions 15.3.1 Second-Order Probability Distribution 15.4 Ensemble Averages, Mean and Autocorrelation 15.5 Stationary Process, Ergodic Process and Temporal Averages 15.5.1 Stationary Process 15.5.2 Temporal Statistics and Ergodic Hypothesis 15.6 Power Spectral Density 15.7 Relationship Between Autocorrelation Function and Power Spectral Density 15.8 Random Response of SDF Systems 15.8.1 Time Domain Analysis 15.8.2 Frequency Domain Analysis 15.9 Random Response of MDF Systems 15.9.1 Complex Matrix Inversion Method 15.9.2 Normal Mode Method 15.10 Response of Flexural Beams Under Random Loading 15.11 Finite Element Random Response of Plates 15.11.1 Cross-Spectral Density Matrix for Generalised Forces 15.11.2 Response Spectra for Displacements and Stresses References 16 Computer Programs in Vibration Analysis 16.1 Introduction 16.2 Computer Program for Forced Vibration Analysis 16.3 Computer Program for Random Vibration Analysis 16.4 Computer Program for Free Vibration Analysis of Framed Structures 16.4.1 A Plane Frame Problem 16.5 Computer Program for the Free Vibration Analysis of Ships by Flexibility Matrix Method 16.6 Computer Program for Finite Element Free Vibration Analysis of Plates 16.6.1 A Plate Problem References Appendix A The Stiffness Matrix Stiffness Matrix Direct Stiffness Method Overall Stiffness Matrix Appendix B Table of Spring Stiffness Index