Dynamics of structures / Jagmohan L. Humar

"A Balkema Book."

Includes bibliographical references and index

Contents Preface Preface to Second Edition List of symbols 1. Introduction 1.1. Objectives of the study of structural dynamics 1.2. Importance of vibration analysis 1.3. Nature of exciting forces 1.3.1. Dynamic forces caused by rotating machinery 1.3.2. Wind loads 1.3.3. Blast loads 1.3.4. Dynamic forces caused by earthquakes 1.3.5. Periodic and nonperiodic loads 1.3.6. Deterministic and nondeterministic loads 1.4. Mathematical modeling of dynamic systems 1.5. Systems of units 1.6. Organization of the text PART 1 2. Formulation of the equations of motion: Single-degree-of-freedom systems 2.1. Introduction 2.2. Inertia forces 2.3. Resultants of inertia forces on a rigid body 2.4. Spring forces 2.5. Damping forces 2.6. Principle of virtual displacement 2.7. Formulation of the equations of motion 2.7.1. Systems with localized mass and localized stiffness 2.7.2. Systems with localized mass but distributed stiffness 2.7.3. Systems with distributed mass but localized stiffness 2.7.4. Systems with distributed stiffness and distributed mass 2.8. Modeling of multi-degree-of-freedom discrete parameter system 2.9. Effect of gravity load 2.10. Axial force effect 2.11. Effect of support motion Selected readings Problems 3. Formulation of the equations of motion: Multi-degree-of-freedom systems 3.1. Introduction 3.2. Principal forces in multi-degree-of-freedom dynamic system 3.2.1. Inertia forces 3.2.2. Forces arising due to elasticity 3.2.3. Damping forces 3.2.4. Axial force effects 3.3. Formulation of the equations of motion 3.3.1. Systems with localized mass and localized stiffness 3.3.2. Systems with localized mass but distributed stiffness 3.3.3. Systems with distributed mass but localized stiffness 3.3.4. Systems with distributed mass and distributed stiffness 3.4. Transformation of coordinates 3.5. Static condensation of stiffness matrix 3.6. Application of Ritz method to discrete systems Selected readings Problems 4. Principles of analytical mechanics 4.1. Introduction 4.2. Generalized coordinates 4.3. Constraints 4.4. Virtual work 4.5. Generalized forces 4.6. Conservative forces and potential energy 4.7. Work function 4.8. Lagrangian multipliers 4.9. Virtual work equation for dynamical systems 4.10. Hamilton's equation 4.11. Lagrange's equation 4.12. Constraint conditions and Lagrangian multipliers 4.13. Lagrange's equations for multi-degree-of-freedom systems 4.14. Rayleigh's dissipation function Selected readings Problems PART 2 5. Free vibration response: Single-degree-of-freedom system 5.1. Introduction 5.2. Undamped free vibration 5.2.1. Phase plane diagram 5.3. Free vibrations with viscous damping 5.3.1. Critically damped system 5.3.2. Overdamped system 5.3.3. Underdamped system 5.3.4. Phase plane diagram 5.3.5. Logarithmic decrement 5.4. Damped free vibration with hysteretic damping 5.5. Damped free vibration with coulomb damping 5.5.1. Phase plane representation of vibrations under Coulomb damping Selected readings Problems 6. Forced harmonic vibrations: Single-degree-of-freedom system 6.1. Introduction 6.2. Procedures for the solution of the forced vibration equation 6.3. Undamped harmonic vibration 6.4. Resonant response of an undamped system 6.5. Damped harmonic vibration 6.6. Complex frequency response 6.7. Resonant response of a damped system 6.8. Rotating unbalanced force 6.9. Transmitted motion due to support movement 6.10. Transmissibility and vibration isolation 6.11. Vibration measuring instruments 6.11.1. Measurement of support acceleration 6.11.2. Measurement of support displacement 6.12. Energy dissipated in viscous damping 6.13. Hysteretic damping 6.14. Complex stiffness 6.15. Coulomb damping 6.16. Measurement of damping 6.16.1. Free vibration decay 6.16.2. Forced-vibration response Selected readings Problems 7. Response to general dynamic loading and transient response 7.1. Introduction 7.2. Response to an Impulsive Force 7.3. Response to general dynamic loading 7.4. Response to a step function load 7.5. Response to a ramp function load 7.6. Response to a step function load with rise time 7.7. Response to shock loading 7.7.1. Rectangular pulse 7.7.2. Triangular pulse 7.7.3. Sinusoidal pulse 7.7.4. Effect of viscous damping 7.7.5. Approximate response analysis for short-duration pulses 7.8. Response to ground motion 7.8.1. Response to a short-duration ground motion pulse 7.9. Analysis of response by the phase plane diagram Selected readings Problems 8. Analysis of single-degree-of-freedom systems: Approximate and numerical methods 8.1. Introduction 8.2. Conservation of energy 8.3. Application of Rayleigh method to multi-degree-of-freedom systems 8.3.1. Flexural vibrations of a beam 8.4. Improved Rayleigh method 8.5. Selection of an appropriate vibration shape 8.6. Systems with distributed mass and stiffness: analysis of internal forces 8.7. Numerical evaluation of Duhamel's integral 8.7.1. Rectangular summation 8.7.2. Trapezoidal method 8.7.3. Simpson's method 8.8. Direct integration of the equations of motion 8.9. Integration based on piece-wise linear representation of the excitation 8.10. Derivation of general formulas 8.11. Constant-acceleration method 8.12. Newmark's β method 8.12.1. Average acceleration method 8.12.2. Linear acceleration method 8.13. Wilson-method 8.14. Methods based on difference expressions 8.14.1. Central difference method 8.14.2. Houbolt's method 8.15. Errors involved in numerical integration 8.16. Stability of the integration method 8.16.1. Newmark's β method 8.16.2. Wilson-method 8.16.3. Central difference method 8.16.4. Houbolt's method 8.17. Selection of a numerical integration method 8.18. Selection of time step Selected readings Problems 9. Analysis of response in the frequency domain 9.1. Transform methods of analysis 9.2. Fourier series representation of a periodic function 9.3. Response to a periodically applied load 9.4. Exponential form of Fourier series 9.5. Complex frequency response function 9.6. Fourier integral representation of a nonperiodic load 9.7. Response to a nonperiodic load 9.8. Convolution integral and convolution theorem 9.9. Discrete Fourier transform 9.10. Discrete convolution and discrete convolution theorem 9.11. Comparison of continuous and discrete fourier transforms 9.12. Application of discrete inverse transform 9.13. Comparison between continuous and discrete convolution 9.14. Discrete convolution of an infinite- and a finite-duration waveform 9.15. Corrective response superposition methods 9.15.1. Corrective transient response based on initial conditions 9.15.2. Corrective periodic response based on initial conditions 9.15.3. Corrective responses obtained from a pair of force pulses 9.16. Exponential window method 9.17. The fast Fourier transform 9.18. Theoretical background to fast Fourier transform 9.19. Computing speed of FFT convolution Selected readings Problems PART 3 10. Free vibration response: Multi-degree-of-freedom system 10.1. Introduction 10.2. Standard eigenvalue problem 10.3. Linearized eigenvalue problem and its properties 10.4. Expansion theorem 10.5. Rayleigh quotient 10.6. Solution of the undamped free vibration problem 10.7. Mode superposition analysis of free-vibration response 10.8. Solution of the damped free-vibration problem 10.9. Additional orthogonality conditions 10.10. Damping orthogonality Selected readings Problems 11. Numerical solution of the eigenproblem 11.1. Introduction 11.2. Properties of standard eigenvalues and eigenvectors 11.3. Transformation of a linearized eigenvalue

problem to the standard form 11.4. Transformation methods 11.4.1. Jacobi diagonalization 11.4.2. Householder's transformation 11.4.3. QR transformation 11.5. Iteration methods 11.5.1. Vector iteration 11.5.2. Inverse vector iteration 11.5.3. Vector iteration with shifts 11.5.4. Subspace iteration 11.5.5. Lanczos iteration 11.6. Determinant search method 11.7. Numerical solution of complex eigenvalue problem 11.7.1. Eigenvalue problem and the orthogonality relationship 11.7.2. Matrix iteration for determining the complex eigenvalues 11.8. Semidefinite or unrestrained systems 11.8.1. Characteristics of an unrestrained system 11.8.2. Eigenvalue solution of a semidefinite system 11.9. Selection of a method for the determination of eigenvalues Selected readings Problems 12. Forced dynamic response: Multi-degree-of-freedom systems 12.1. Introduction 12.2. Normal coordinate transformation 12.3. Summary of mode superposition method 12.4. Complex frequency response 12.5. Vibration absorbers 12.6. Effect of support excitation 12.7. Forced vibration of unrestrained system Selected readings Problems 13. Analysis of multi-degree-of-freedom systems: Approximate and numerical methods 13.1. Introduction 13.2. Rayleigh-Ritz method 13.3. Application of Ritz method to forced vibration response 13.3.1. Mode superposition method 13.3.2. Mode acceleration method 13.3.3. Static condensation and Guyan's reduction 13.3.4. Load-dependent Ritz vectors 13.3.5. Application of lanczos vectors in the transformation of the equations of motion 13.4. Direct integration of the equations of motion 13.4.1. Explicit integration schemes 13.4.2. Implicit integration schemes 13.4.3. Mixed methods in direct integration 13.5. Analysis in the frequency domain 13.5.1. Frequency analysis of systems with classical mode shapes 13.5.2. Frequency analysis of systems without classical mode shapes Selected readings Problems PART 4 14. Formulation of the equations of motion: Continuous systems 14.1. Introduction 14.2. Transverse vibrations of a beam 14.3. Transverse vibrations of a beam: variational formulation 14.4. Effect of damping resistance on transverse vibrations of a beam 14.5. Effect of shear deformation and rotatory inertia on the flexural vibrations of a beam 14.6. Axial vibrations of a bar 14.7. Torsional vibrations of a bar 14.8. Transverse vibrations of a string 14.9. Transverse vibrations of a shear beam 14.10. Transverse vibrations of a beam excited by support motion 14.11. Effect of axial force on transverse vibrations of a beam Selected readings Problems 15. Continuous systems: Free vibration response 15.1. Introduction 15.2. Eigenvalue problem for the transverse vibrations of a beam 15.3. General eigenvalue problem for a continuous system 15.3.1. Definition of the eigenvalue problem 15.3.2. Self-adjointness of operators in the eigenvalue problem 15.3.3. Orthogonality of eigenfunctions 15.3.4. Positive and positive definite operators 15.4. Expansion theorem 15.5. Frequencies and mode shapes for lateral vibrations of a beam 15.5.1. Simply supported beam 15.5.2. Uniform cantilever beam 15.5.3. Uniform beam clamped at both ends 15.5.4. Uniform beam with both ends free 15.6. Effect of shear deformation and rotatory inertia on the frequencies of flexural vibrations 15.7. Frequencies and mode shapes for the axial vibrations of a bar 15.7.1. Axial vibrations of a clamped-free bar 15.7.2. Axial vibrations of a free-free bar 15.8. Frequencies and mode shapes for the transverse vibration of a string 15.8.1. Vibrations of a string tied at both ends 15.9. Boundary conditions containing the eigenvalue 15.10. Free-vibration response of a continuous system 15.11. Undamped free transverse vibrations of a beam 15.12. Damped free transverse vibrations of a beam Selected readings Problems 16. Continuous systems: Forced-vibration response 16.1. Introduction 16.2. Normal coordinate transformation: general case of an undamped system 16.3. Forced lateral vibration of a beam 16.4. Transverse vibrations of a beam under traveling load 16.5. Forced axial vibrations of a uniform bar 16.6. Normal coordinate transformation, damped case Selected readings Problems 17. Wave propagation analysis 17.1. Introduction 17.2. The Phenomenon of wave propagation 17.3. Harmonic waves 17.4. One dimensional wave equation and its solution 17.5. Propagation of waves in systems of finite extent 17.6. Reflection and refraction of waves at a discontinuity in the system properties 17.7. Characteristics of the wave equation 17.8. Wave dispersion Selected readings Problems PART 5 18. Finite element method 18.1. Introduction 18.2. Formulation of the finite element equations 18.3. Selection of shape functions 18.4. Advantages of the finite element method 18.5. Element Shapes 18.5.1. One-dimensional elements 18.5.2. Two-dimensional elements 18.6. One-dimensional bar element 18.7. Flexural vibrations of a beam 18.7.1. Stiffness matrix of a beam element 18.7.2. Mass matrix of a beam element 18.7.3. Nodal applied force vector for a beam element 18.7.4. Geometric stiffness matrix for a beam element 18.7.5. Simultaneous axial and lateral vibrations 18.8. Stress-strain relationships for a continuum 18.8.1. Plane stress 18.8.2. Plane strain 18.9. Triangular element in plane stress and plane strain 18.10. Natural coordinates 18.10.1. Natural coordinate formulation for a uniaxial bar element 18.10.2. Natural coordinate formulation for a constant strain triangle 18.10.3. Natural coordinate formulation for a linear strain triangle Selected readings Problems 19. Component mode synthesis 19.1. Introduction 19.2. Fixed interface methods 19.2.1. Fixed interface normal modes 19.2.2. Constraint modes 19.2.3. Transformation of coordinates 19.2.4. Illustrative example 19.3. Free interface method 19.3.1. Free interface normal modes 19.3.2. Attachment modes 19.3.3. Inertia relief attachment modes 19.3.4. Residual flexibility attachment modes 19.3.5. Transformation of coordinates 19.3.6. Illustrative example 19.4. Hybrid method 19.4.1. Experimental determination of modal parameters 19.4.2. Experimental determination of the static constraint modes 19.4.3. Component modes and transformation of component matrices 19.4.4. Illustrative example Selected readings Problems 20. Analysis of nonlinear response 20.1. Introduction 20.2. Single-degree-of freedom system 20.2.1. Central difference method 20.2.2. Newmark's β Method 20.3. Errors involved in numerical integration of nonlinear systems 20.4. Multiple degree-of-freedom system 20.4.1. Explicit integration 20.4.2. Implicit integration 20.4.3. Iterations within a time step Selected readings Problems Answers to selected problems Index