Table of Contents. Preface... 13

Transcription

1 Table of Contents Preface Chapter 1. Vibrations of Continuous Elastic Solid Media Objective of the chapter Equations of motion and boundary conditions of continuous media Description of the movement of continuous media Law of conservation Conservation of mass Conservation of momentum Conservation of energy Boundary conditions Study of the vibrations: small movements around a position of static, stable equilibrium Linearization around a configuration of reference Elastic solid continuous media Summary of the problem of small movements of an elastic continuous medium in adiabatic mode Position of static equilibrium of an elastic solid medium Vibrations of elastic solid media Boundary conditions Vibrations equations Notes on the initial conditions of the problem of vibrations Formulation in displacement Vibration of viscoelastic solid media Conclusion... 44

2 6 Vibration in Continuous Media Chapter 2. Variational Formulation for Vibrations of Elastic Continuous Media Objective of the chapter Concept of the functional, bases of the variational method The problem Fundamental lemma Basis of variational formulation Directional derivative Extremum of a functional calculus Reissner s functional Basic functional Some particular cases of boundary conditions Case of boundary conditions effects of rigidity and mass Hamilton s functional The basic functional Some particular cases of boundary conditions Approximate solutions Euler equations associated to the extremum of a functional Introduction and first example Second example: vibrations of plates Some results Conclusion Chapter 3. Equation of Motion for Beams Objective of the chapter Hypotheses of condensation of straight beams Equations of longitudinal vibrations of straight beams Basic equations with mixed variables Equations with displacement variables Equations with displacement variables obtained by Hamilton s functional Equations of vibrations of torsion of straight beams Basic equations with mixed variables Equation with displacements Equations of bending vibrations of straight beams Basic equations with mixed variables: Timoshenko s beam Equations with displacement variables: Timoshenko s beam Basic equations with mixed variables: Euler-Bernoulli beam Equations of the Euler-Bernoulli beam with displacement variable Complex vibratory movements: sandwich beam with a flexible inside Conclusion...109

3 Table of Contents 7 Chapter 4. Equation of Vibration for Plates Objective of the chapter Thin plate hypotheses General procedure In plane vibrations Transverse vibrations: Mindlin s hypotheses Transverse vibrations: Love-Kirchhoff hypotheses Plates which are non-homogenous in thickness Equations of motion and boundary conditions of in plane vibrations Equations of motion and boundary conditions of transverse vibrations Mindlin s hypotheses: equations with mixed variables Mindlin s hypotheses: equations with displacement variables Love-Kirchhoff hypotheses: equations with mixed variables Love-Kirchhoff hypotheses: equations with displacement variables Love-Kirchhoff hypotheses: equations with displacement variables obtained using Hamilton s functional Some comments on the formulations of transverse vibrations Coupled movements Equations with polar co-ordinates Basic relations Love-Kirchhoff equations of the transverse vibrations of plates Conclusion Chapter 5. Vibratory Phenomena Described by the Wave Equation Introduction Wave equation: presentation of the problem and uniqueness of the solution The wave equation Equation of energy and uniqueness of the solution Resolution of the wave equation by the method of propagation (d Alembert s methodology) General solution of the wave equation Taking initial conditions into account Taking into account boundary conditions: image source Resolution of the wave equation by separation of variables General solution of the wave equation in the form of separate variables Taking boundary conditions into account Taking initial conditions into account Orthogonality of mode shapes...165

4 8 Vibration in Continuous Media 5.5. Applications Longitudinal vibrations of a clamped-free beam Torsion vibrations of a line of shafts with a reducer Conclusion Chapter 6. Free Bending Vibration of Beams Introduction The problem Solution of the equation of the homogenous beam with a constant cross-section Solution Interpretation of the vibratory solution, traveling waves, vanishing waves Propagation in infinite beams Introduction Propagation of a group of waves Introduction of boundary conditions: vibration modes Introduction The case of the supported-supported beam The case of the supported-clamped beam The free-free beam Summary table Stress-displacement connection Influence of secondary effects Influence of rotational inertia Influence of transverse shearing Taking into account shearing and rotational inertia Conclusion Chapter 7. Bending Vibration of Plates Introduction Posing the problem: writing down boundary conditions Solution of the equation of motion by separation of variables Separation of the space and time variables Solution of the equation of motion by separation of space variables Solution of the equation of motion (second method) Vibration modes of plates supported at two opposite edges General case Plate supported at its four edges Physical interpretation of the vibration modes...244

5 Table of Contents The particular case of square plates Second method of calculation Vibration modes of rectangular plates: approximation by the edge effect method General issues Formulation of the method The plate clamped at its four edges Another type of boundary conditions Approximation of the mode shapes Calculation of the free vibratory response following the application of initial conditions Circular plates Equation of motion and solution by separation of variables Vibration modes of the full circular plate clamped at the edge Modal system of a ring-shaped plate Conclusion Chapter 8. Introduction to Damping: Example of the Wave Equation Introduction Wave equation with viscous damping Damping by dissipative boundary conditions Presentation of the problem Solution of the problem Calculation of the vibratory response Viscoelastic beam Properties of orthogonality of damped systems Conclusion Chapter 9. Calculation of Forced Vibrations by Modal Expansion Objective of the chapter Stages of the calculation of response by modal decomposition Reference example Overview Taking damping into account Examples of calculation of generalized mass and stiffness Homogenous, isotropic beam in pure bending Isotropic homogenous beam in pure bending with a rotational inertia effect Solution of the modal equation Solution of the modal equation for a harmonic excitation Solution of the modal equation for an impulse excitation...330

6 10 Vibration in Continuous Media Unspecified excitation, solution in frequency domain Unspecified excitation, solution in time domain Example response calculation Response of a bending beam excited by a harmonic force Response of a beam in longitudinal vibration excited by an impulse force (time domain calculation) Response of a beam in longitudinal vibrations subjected to an impulse force (frequency domain calculation) Convergence of modal series Convergence of modal series in the case of harmonic excitations Acceleration of the convergence of modal series of forced harmonic responses Conclusion Chapter 10. Calculation of Forced Vibrations by Forced Wave Decomposition Introduction Introduction to the method on the example of a beam in torsion Example: homogenous beam in torsion Forced waves Calculation of the forced response Heterogenous beam Excitation by imposed displacement Resolution of the problems of bending Example of an excitation by force Excitation by torque Damped media (case of the longitudinal vibrations of beams) Example Generalization: distributed excitations and non-harmonic excitations Distributed excitations Non-harmonic excitations Unspecified homogenous mono-dimensional medium Forced vibrations of rectangular plates Conclusion Chapter 11. The Rayleigh-Ritz Method based on Reissner s Functional Introduction Variational formulation of the vibrations of bending of beams Generation of functional spaces Approximation of the vibratory response Formulation of the method...392

7 Table of Contents Application to the vibrations of a clamped-free beam Construction of a polynomial base Modeling with one degree of freedom Model with two degrees of freedom Model with one degree of freedom verifying the displacement and stress boundary conditions Conclusion Chapter 12. The Rayleigh-Ritz Method based on Hamilton s Functional Introduction Reference example: bending vibrations of beams Hamilton s variational formulation Formulation of the Rayleigh-Ritz method Application: use of a polynomial base for the clamped-free beam Functional base of the finite elements type: application to longitudinal vibrations of beams Functional base of the modal type: application to plates equipped with heterogenities Elastic boundary conditions Introduction The problem Approximation with two terms Convergence of the Rayleigh-Ritz method Introduction The Rayleigh quotient Introduction to the modal system as an extremum of the Rayleigh quotient Approximation of the normal angular frequencies by the Rayleigh quotient or the Rayleigh-Ritz method Conclusion Bibliography and Further Reading Index...439