Vibration Mechanics

1 Introduction.- 1.1 Brief history.- 1.2 Disruptive or useful vibrations.- 2 The Linear Elementary Oscillator of Mechanics.- 2.1 Definitions and notation.- 2.2 Equation of motion and vibratory states.- 2.3 Modified forms of the equation of motion.- 3 The Free State of the Elementary Oscillator.- 3.1 Conservative free state * Harmonic oscillator.- 3.2 Conservation of energy.- 3.3 Examples of conservative oscillators.- 3.3.1 Introduction.- 3.3.2 Mass at the end of a wire.- 3.3.3 Lateral vibrations of a shaft.- 3.3.4 Pendulum system.- 3.3.5 Helmholtz resonator.- 3.4 Dissipative free state.- 3.4.1 Super-critical damping.- 3.4.2 Critical damping.- 3.4.3 Sub-critical damping.- 3.5 Energy of the dissipative oscillator.- 3.6 Phase plane graph.- 3.7 Examples of dissipative oscillators.- 3.7.1 Suspension element for a vehicle.- 3.7.2 Damping of a polymer bar.- 3.7.3 Oscillator with dry friction.- 4 Harmonic Steady State.- 4.1 Amplitude and phase as a function of frequency.- 4.2 Graph of rotating vectors.- 4.3 Use of complex numbers * Frequency response.- 4.4 Power consumed in the steady state.- 4.5 Natural and resonant angular frequencies.- 4.6 The Nyquist graph.- 4.7 Examples of harmonic steady states.- 4.7.1 Vibrator for fatigue tests.- 4.7.2 Measurement of damping.- 4.7.3 Vibrations of a machine shaft.- 5 Periodic Steady State.- 5.1 Fourier series * Excitation and response spectra.- 5.2 Complex form of the Fourier series.- 5.3 Examples of periodic steady states.- 5.3.1 Steady state beats.- 5.3.2 Response to a periodic rectangular excitation.- 5.3.3 Time response to a periodic excitation.- 6 Forced State.- 6.1 Laplace transform.- 6.2 General solution of the forced state.- 6.3 Response to an impulse and to a unit step force.- 6.3.1 Impulse response.- 6.3.2 Indicial response.- 6.3.3 Relation between the impulse and indicial responses.- 6.4 Responses to an impulse and to a unit step elastic displacement.- 6.4.1 Introduction.- 6.4.2 Impulse response.- 6.4.3 Indicial response.- 6.5 Fourier transformation.- 6.6 Examples of forced states.- 6.6.1 Time response to a force F cos ?t.- 6.6.2 Frequency response to a rectangular excitation.- 7 Electrical Analogues.- 7.1 Generalities.- 7.2 Force-current analogy.- 7.3 Extension to systems with several degrees of freedom * Circuits of forces.- 8 Systems with Two Degrees of Freedom.- 8.1 Generalities * Concept of coupling.- 8.2 Free state and natural modes of the conservative system.- 8.3 Study of elastic coupling.- 8.4 Examples of oscillators with two degrees of freedom.- 8.4.1 Natural frequencies of a service lift.- 8.4.2 Beats in the free state.- 9 The Frahm Damper.- 9.1 Definition and differential equations of the system.- 9.2 Harmonic steady state.- 9.3 Limiting cases of the damping.- 9.4 Optimization of the Frahm damper.- 9.5 Examples of applications.- 9.6 The Lanchester damper.- 10 The Concept of the Generalized Oscillator.- 10.1 Definition and energetic forms of the generalized oscillator.- 10.2 Differentiation of a symmetrical quadratic form * Equations of Lagrange.- 10.3 Examination of particular cases.- 10.3.1 Energetic forms of the oscillator with two degrees of freedom.- 10.3.2 Potential energy of a linear elastic system.- 10.3.3 Kinetic energy of a system of point masses.- 11 Free State of the Conservative Generalized Oscillator.- 11.1 Introduction.- 11.2 Solution of the system by linear combination of specific solutions.- 11.2.1 Search for specific solutions.- 11.2.2 General solution * Natural modes.- 11.2.3 Other forms of the characteristic equation.- 11.2.4 Summary and comments * Additional constraints.- 11.3 Solution of the system by change of coordinates.- 11.3.1 Decoupling of the equations * Normal coordinates.- 11.3.2 Eigenvalue problem.- 11.3.3 Energetic forms * Sign of the eigenvalues.- 11.3.4 General form of the solution.- 11.3.5 Linear independence and orthogonality of the modal vectors.- 11.3.6 Normalization of the natural mode shapes.- 11.4 Response to an initial excitation.- 11.5 Rayleigh quotient.- 11.6 Examples of conservative generalized oscillators.- 11.6.1 Symmetrical triple pendulum.- 11.6.2 Masses concentrated along a cord.- 11.6.3 Masses concentrated along a beam.- 11.6.4 Study of the behaviour of a milling table.- 12 Free State of the Dissipative Generalized Oscillator.- 12.1 Limits of classical modal analysis.- 12.2 Dissipative free state with real modes.- 12.3 Response to an initial excitation in the case of real modes.- 12.4 General case.- 12.5 Hamiltonian equations for the system.- 12.6 Solution of the differential system.- 12.6.1 Change of coordinates * Phase space.- 12.6.2 Eigenvalue problem.- 12.6.3 General solution.- 12.6.4 Orthogonality of the modal vectors * Normalization.- 12.7 Response to an initial excitation in the general case.- 12.8 Direct search for specific solutions.- 12.9 Another form of the characteristic equation.- 13 Example of Visualization of Complex Natural Modes.- 13.1 Description of the system.- 13.2 Energetic form * Differential equation.- 13.3 Isolation of a mode.- 13.3.1 General case.- 13.3.2 Principal axes of the trajectory.- 13.3.3 Conservative system.- 13.4 Numerical examples.- 13.4.1 Equations of motion.- 13.4.2 Isolation of the first mode.- 13.4.3 Isolation of the second mode.- 13.4.4 Conservative system.- 13.5 Summary and comments.- 14 Forced State of the Generalized Oscillator.- 14.1 Introduction.- 14.2 Dissipative systems with real modes.- 14.3 Dissipative systems in the general case.- 14.4 Introduction to experimental modal analysis.- Symbol List.