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Objectives This book is used to teach vibratory mechanics to undergraduate engineers at the Swiss Federal Institute of Technology of Lausanne. It is a basic course, at the level of the first university degree, necessary for the proper comprehension of the following disciplines. Vibrations of continuous linear systems (beams, plates) random vibration of linear systems vibrations of non-linear systems dynamics of structures experimental methods, rheological models, etc. Effective teaching methods have been given the highest priority. Thus the book covers basic theories of vibratory mechanics in an ap propriately rigorous and complete way, and is illustrated by nume rous applied examples. In addition to university students, it is suitable for industrial engineers who want to strengthen or complete their training. It has been written so that someone working alone should find it easy to read. pescription The subject of the book is the vibrations of linear mechanical sys tems having only a finite number of degrees of freedom (ie discrete linear systems). These can be divided into the following two catego ries : -X- systems of solids which are considered to be rigid, and which are acted upon by elastic forces and by linear resist.ive forces (viscous damping forces). deformable continuous systems which have been made discrete. In other words, systems which are replaced (approximately) by systems having only a limited number of degrees of freedom, using digital or experimental methods.

### Details & Specs

Title:Vibration Mechanics: Linear Discrete SystemsFormat:PaperbackDimensions:335 pages, 23.5 × 15.5 × 0.02 inPublished:September 21, 2012Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9401055548

ISBN - 13:9789401055543

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### Extra Content

Table of Contents

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.