Vibration Mechanics

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.