Advanced Mechanical Vibrations Physics, Mathematics and Applications

Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Preface
Acknowledgements
Frequently Used Acronyms
1 A Few Preliminary Fundamentals
	1.1 Introduction
	1.2 Modelling Vibrations and Vibrating Systems
	1.3 Some Basic Concepts
		1.3.1 The Phenomenon of Beats
		1.3.2 Displacement, Velocity, Acceleration and Decibels
	1.4 Springs, Dampers and Masses
2 Formulating the Equations Of Motion
	2.1 Introduction
	2.2 Systems of Material Particles
		2.2.1 Generalised Co-Ordinates, Constraints and Degrees of Freedom
	2.3 Virtual Work and d’Ahlembert’s Principles – Lagrange and Hamilton Equations
		2.3.1 Hamilton’s Equations (HEs)
	2.4 On the Properties and Structure of Lagrange’s Equations
		2.4.1 Invariance in the Form of Les and Monogenic Forces
		2.4.2 The Structure of the Kinetic Energy and of Lagrange Equations
		2.4.3 The Energy Function and the Conservation of Energy
		2.4.4 Elastic Forces, Viscous Forces and Rayleigh Dissipation Function
		2.4.5 More Co-Ordinates than DOFs: Lagrange’s Multipliers
	2.5 Hamilton’s Principle
		2.5.1 More than One Independent Variable: Continuous Systems and Boundary Conditions
	2.6 Small-Amplitude Oscillations
	2.7 A Few Complements
		2.7.1 Motion in a Non-Inertial Frame of Reference
		2.7.2 Uniformly Rotating Frame
		2.7.3 Ignorable Co-Ordinates and the Routh Function
		2.7.4 The Simple Pendulum Again: A Note on Non-Small Oscillations
3 Finite DOFs Systems: Free Vibration
	3.1 Introduction
	3.2 Free Vibration of 1-DOF Systems
		3.2.1 Logarithmic Decrement
	3.3 Free Vibration of MDOF Systems: The Undamped Case
		3.3.1 Orthogonality of Eigenvectors and Normalisation
		3.3.2 The General Solution of the Undamped Free-Vibration Problem
		3.3.3 Normal Co-Ordinates
		3.3.4 Eigenvalues and Eigenvectors Sensitivities
		3.3.5 Light Damping as a Perturbation of an Undamped System
		3.3.6 More Orthogonality Conditions
		3.3.7 Eigenvalue Degeneracy
		3.3.8 Unrestrained Systems: Rigid-Body Modes
	3.4 Damped Systems: Classical and Non-Classical Damping
		3.4.1 Rayleigh Damping
		3.4.2 Non-Classical Damping
	3.5 GEPs and QEPs: Reduction to Standard Form
		3.5.1 Undamped Systems
		3.5.2 Viscously Damped Systems
	3.6 Eigenvalues Sensitivity of Viscously Damped Systems
4 Finite-DOFs Systems: Response to External Excitation
	4.1 Introduction
	4.2 Response in the Time-, Frequency- and S-Domains: IRF, Duhamel’s Integral, FRF and TF
		4.2.1 Excitation Due to Base Displacement, Velocity or Acceleration
	4.3 Harmonic and Periodic Excitation
		4.3.1 A Few Notes on Vibration Isolation
		4.3.2 Eccentric Excitation
		4.3.3 Other Forms of FRFs
		4.3.4 Damping Evaluation
		4.3.5 Response Spectrum
	4.4 MDOF Systems: Classical Damping
		4.4.1 Mode ‘Truncation’ and the Mode-Acceleration Solution
		4.4.2 The Presence of Rigid-Body Modes
	4.5 MDOF Systems: Non-Classical Viscous Damping, a State-Space Approach
		4.5.1 Another State-Space Formulation
	4.6 Frequency Response Functions of a 2-DOF System
	4.7 A Few Further Remarks on FRFs
5 Vibrations of Continuous Systems
	5.1 Introduction
	5.2 The Flexible String
		5.2.1 Sinusoidal Waveforms and Standing Waves
		5.2.2 Finite Strings: The Presence of Boundaries and the Free Vibration
	5.3 Free Longitudinal and Torsional Vibration of Bars
	5.4 A Short Mathematical Interlude: Sturm–Liouville Problems
	5.5 A Two-Dimensional System: Free Vibration of a Flexible Membrane
		5.5.1 Circular Membrane with Fixed Edge
	5.6 Flexural (Bending) Vibrations of Beams
	5.7 Finite Beams With Classical BCs
		5.7.1 On the Orthogonality of Beam Eigenfunctions
		5.7.2 Axial Force Effects
		5.7.3 Shear Deformation and Rotary Inertia (Timoshenko Beam)
	5.8 Bending Vibrations of Thin Plates
		5.8.1 Rectangular Plates
		5.8.2 Circular Plates
		5.8.3 On the Orthogonality of Plate Eigenfunctions
	5.9 A Few Additional Remarks
		5.9.1 Self-Adjointness and Positive-Definiteness of the Beam and Plate Operators
		5.9.2 Analogy with Finite-DOFs Systems
		5.9.3 The Free Vibration Solution
	5.10 Forced Vibrations: The Modal Approach
		5.10.1 Alternative Closed-Form for FRFs
		5.10.2 A Note on Green’s Functions
6 Random Vibrations
	6.1 Introduction
	6.2 The Concept of Random Process, Correlation and Covariance Functions
		6.2.1 Stationary Processes
		6.2.2 Main Properties of Correlation and Covariance Functions
		6.2.3 Ergodic Processes
	6.3 Some Calculus for Random Processes
	6.4 Spectral Representation of Stationary Random Processes
		6.4.1 Main Properties of Spectral Densities
		6.4.2 Narrowband and Broadband Processes
	6.5 Response of Linear Systems to Stationary Random Excitation
		6.5.1 SISO (Single Input–Single Output) Systems
		6.5.2 SDOF-System Response to Broadband Excitation
		6.5.3 SDOF Systems: Transient Response
		6.5.4 A Note on Gaussian (Normal) Processes
		6.5.5 MIMO (Multiple Inputs–Multiple Outputs) Systems
		6.5.6 Response of MDOF Systems
		6.5.7 Response of a Continuous System to Distributed Random Excitation: A Modal Approach
	6.6 Threshold Crossing Rates and Peaks Distribution of Stationary Narrowband Processes
Appendix A: On Matrices and Linear Spaces
Appendix B: Fourier Series, Fourier and Laplace Transforms
References and Further Reading
Index