Vibration in Continuous Media

Jean-Louis Guyader is Professor of Vibration and Acoustics within the Mechanical Engineering Department at INSA, Lyon, France and Director of the Vibration and Acoustics Laboratory. His research covers the acoustic radiation of structures in light or heavy fluids and the energy propagation in vibrating structures and acoustic media.

• Preface 13Chapter 1. Vibrations of Continuous Elastic Solid Media 171.1. Objective of the chapter 171.2. Equations of motion and boundary conditions of continuous media 181.2.1. Description of the movement of continuous media 181.2.2. Law of conservation 211.2.3. Conservation of mass 231.2.4. Conservation of momentum 231.2.5. Conservation of energy 251.2.6. Boundary conditions 261.3. Study of the vibrations: small movements around a position of static, stable equilibrium 281.3.1. Linearization around a configuration of reference 281.3.2. Elastic solid continuous media 321.3.3. Summary of the problem of small movements of an elastic continuous medium in adiabatic mode 331.3.4. Position of static equilibrium of an elastic solid medium 341.3.5. Vibrations of elastic solid media 351.3.6. Boundary conditions 371.3.7. Vibrations equations 381.3.8. Notes on the initial conditions of the problem of vibrations 391.3.9. Formulation in displacement 401.3.10. Vibration of viscoelastic solid media 401.4. Conclusion 44Chapter 2. Variational Formulation for Vibrations of Elastic Continuous Media 452.1. Objective of the chapter 452.2. Concept of the functional, bases of the variational method 462.2.1. The problem 462.2.2. Fundamental lemma 462.2.3. Basis of variational formulation 472.2.4. Directional derivative 502.2.5. Extremum of a functional calculus 552.3. Reissner’s functional 562.3.1. Basic functional 562.3.2. Some particular cases of boundary conditions 592.3.3. Case of boundary conditions effects of rigidity and mass 602.4. Hamilton’s functional 612.4.1. The basic functional 612.4.2. Some particular cases of boundary conditions 622.5. Approximate solutions 632.6. Euler equations associated to the extremum of a functional 642.6.1. Introduction and first example 642.6.2. Second example: vibrations of plates 682.6.3. Some results 722.7. Conclusion 75Chapter 3. Equation of Motion for Beams 773.1. Objective of the chapter 773.2. Hypotheses of condensation of straight beams 783.3. Equations of longitudinal vibrations of straight beams 803.3.1. Basic equations with mixed variables 803.3.2. Equations with displacement variables 853.3.3. Equations with displacement variables obtained by Hamilton’s functional 863.4. Equations of vibrations of torsion of straight beams 893.4.1. Basic equations with mixed variables 893.4.2. Equation with displacements 913.5. Equations of bending vibrations of straight beams 933.5.1. Basic equations with mixed variables: Timoshenko’s beam 933.5.2. Equations with displacement variables: Timoshenko’s beam 973.5.3. Basic equations with mixed variables: Euler-Bernoulli beam 1013.5.4. Equations of the Euler-Bernoulli beam with displacement variable 1023.6. Complex vibratory movements: sandwich beam with a flexible inside 1043.7. Conclusion 109Chapter 4. Equation of Vibration for Plates 1114.1. Objective of the chapter 1114.2. Thin plate hypotheses 1124.2.1. General procedure 1124.2.2. In plane vibrations 1124.2.3. Transverse vibrations: Mindlin’s hypotheses 1134.2.4. Transverse vibrations: Love-Kirchhoff hypotheses 1144.2.5. Plates which are non-homogenous in thickness 1154.3. Equations of motion and boundary conditions of in plane vibrations 1164.4. Equations of motion and boundary conditions of transverse vibrations 1214.4.1. Mindlin’s hypotheses: equations with mixed variables 1214.4.2. Mindlin’s hypotheses: equations with displacement variables 1234.4.3. Love-Kirchhoff hypotheses: equations with mixed variables 1244.4.4. Love-Kirchhoff hypotheses: equations with displacement variables 1274.4.5. Love-Kirchhoff hypotheses: equations with displacement variables obtained using Hamilton’s functional 1294.4.6. Some comments on the formulations of transverse vibrations 1304.5. Coupled movements 1304.6. Equations with polar co-ordinates 1334.6.1. Basic relations 1334.6.2. Love-Kirchhoff equations of the transverse vibrations of plates 1354.7. Conclusion 138Chapter 5. Vibratory Phenomena Described by the Wave Equation 1395.1. Introduction 1395.2. Wave equation: presentation of the problem and uniqueness of the solution 1405.2.1. The wave equation 1405.2.2. Equation of energy and uniqueness of the solution 1425.3. Resolution of the wave equation by the method of propagation (d’Alembert’s methodology) 1455.3.1. General solution of the wave equation 1455.3.2. Taking initial conditions into account 1475.3.3. Taking into account boundary conditions: image source 1515.4. Resolution of the wave equation by separation of variables 1545.4.1. General solution of the wave equation in the form of separate variables 1545.4.2. Taking boundary conditions into account 1575.4.3. Taking initial conditions into account 1635.4.4. Orthogonality of mode shapes 1655.5. Applications 1685.5.1. Longitudinal vibrations of a clamped-free beam 1685.5.2. Torsion vibrations of a line of shafts with a reducer 1725.6. Conclusion 178Chapter 6. Free Bending Vibration of Beams 1816.1. Introduction 1816.2. The problem 1826.3. Solution of the equation of the homogenous beam with a constant cross-section 1846.3.1. Solution 1846.3.2. Interpretation of the vibratory solution, traveling waves, vanishing waves 1866.4. Propagation in infinite beams 1896.4.1. Introduction 1896.4.2. Propagation of a group of waves 1916.5. Introduction of boundary conditions: vibration modes 1976.5.1. Introduction 1976.5.2. The case of the supported-supported beam 1976.5.3. The case of the supported-clamped beam 2016.5.4. The free-free beam 2066.5.5. Summary table 2096.6. Stress-displacement connection 2106.7. Influence of secondary effects 2116.7.1. Influence of rotational inertia 2126.7.2. Influence of transverse shearing 2156.7.3. Taking into account shearing and rotational inertia 2216.8. Conclusion 227Chapter 7. Bending Vibration of Plates 2297.1. Introduction 2297.2. Posing the problem: writing down boundary conditions 2307.3. Solution of the equation of motion by separation of variables 2347.3.1. Separation of the space and time variables 2347.3.2. Solution of the equation of motion by separation of space variables 2357.3.3. Solution of the equation of motion (second method) 2377.4. Vibration modes of plates supported at two opposite edges 2397.4.1. General case 2397.4.2. Plate supported at its four edges 2417.4.3. Physical interpretation of the vibration modes 2447.4.4. The particular case of square plates 2487.4.5. Second method of calculation 2517.5. Vibration modes of rectangular plates: approximation by the edge effect method 2547.5.1. General issues 2547.5.2. Formulation of the method 2557.5.3. The plate clamped at its four edges 2597.5.4. Another type of boundary conditions 2617.5.5. Approximation of the mode shapes 2637.6. Calculation of the free vibratory response following the application of initial conditions 2637.7. Circular plates 2657.7.1. Equation of motion and solution by separation of variables 2657.7.2. Vibration modes of the full circular plate clamped at the edge 2727.7.3. Modal system of a ring-shaped plate 2767.8. Conclusion 277Chapter 8. Introduction to Damping: Example of the Wave Equation 2798.1. Introduction 2798.2. Wave equation with viscous damping 2818.3. Damping by dissipative boundary conditions 2878.3.1. Presentation of the problem 2878.3.2. Solution of the problem 2888.3.3. Calculation of the vibratory response 2948.4. Viscoelastic beam 2978.5. Properties of orthogonality of damped systems 3038.6. Conclusion 308Chapter 9. Calculation of Forced Vibrations by Modal Expansion 3099.1. Objective of the chapter 3099.2. Stages of the calculation of response by modal decomposition 3109.2.1. Reference example 3109.2.2. Overview 3179.2.3. Taking damping into account 3219.3. Examples of calculation of generalized mass and stiffness 3229.3.1. Homogenous, isotropic beam in pure bending 3229.3.2. Isotropic homogenous beam in pure bending with a rotational inertia effect 3239.4. Solution of the modal equation 3249.4.1. Solution of the modal equation for a harmonic excitation 3249.4.2. Solution of the modal equation for an impulse excitation 3309.4.3. Unspecified excitation, solution in frequency domain 3329.4.4. Unspecified excitation, solution in time domain 3339.5. Example response calculation 3369.5.1. Response of a bending beam excited by a harmonic force 3369.5.2. Response of a beam in longitudinal vibration excited by an impulse force (time domain calculation) 3409.5.3. Response of a beam in longitudinal vibrations subjected to an impulse force (frequency domain calculation) 3439.6. Convergence of modal series 3479.6.1. Convergence of modal series in the case of harmonic excitations 3479.6.2. Acceleration of the convergence of modal series of forced harmonic responses 3509.7. Conclusion 353Chapter 10. Calculation of Forced Vibrations by Forced Wave Decomposition 35510.1. Introduction 35510.2. Introduction to the method on the example of a beam in torsion 35610.2.1. Example: homogenous beam in torsion 35610.2.2. Forced waves 35810.2.3. Calculation of the forced response 35910.2.4. Heterogenous beam 36110.2.5. Excitation by imposed displacement 36310.3. Resolution of the problems of bending 36510.3.1. Example of an excitation by force 36510.3.2. Excitation by torque 36810.4. Damped media (case of the longitudinal vibrations of beams) 36910.4.1. Example 36910.5. Generalization: distributed excitations and non-harmonic excitations 37110.5.1. Distributed excitations 37110.5.2. Non-harmonic excitations 37510.5.3. Unspecified homogenous mono-dimensional medium 37710.6 Forced vibrations of rectangular plates 37910.7. Conclusion 385Chapter 11. The Rayleigh-Ritz Method based on Reissner’s Functional 38711.1. Introduction 38711.2. Variational formulation of the vibrations of bending of beams 38811.3. Generation of functional spaces 39111.4. Approximation of the vibratory response 39211.5. Formulation of the method 39211.6. Application to the vibrations of a clamped-free beam 39711.6.1. Construction of a polynomial base 39711.6.2. Modeling with one degree of freedom 39911.6.3. Model with two degrees of freedom 40211.6.4. Model with one degree of freedom verifying the displacement and stress boundary conditions 40411.7. Conclusion 406Chapter 12. The Rayleigh-Ritz Method based on Hamilton’s Functional 40912.1. Introduction 40912.2. Reference example: bending vibrations of beams 40912.2.1 Hamilton’s variational formulation 40912.2.2. Formulation of the Rayleigh-Ritz method 41112.2.3. Application: use of a polynomial base for the clamped-free beam 41412.3. Functional base of the finite elements type: application to longitudinal vibrations of beams 41512.4. Functional base of the modal type: application to plates equipped with heterogenities 42012.5. Elastic boundary conditions 42312.5.1. Introduction 42312.5.2. The problem 42312.5.3. Approximation with two terms 42412.6. Convergence of the Rayleigh-Ritz method 42612.6.1. Introduction 42612.6.2. The Rayleigh quotient 42612.6.3. Introduction to the modal system as an extremum of the Rayleigh quotient 42812.6.4. Approximation of the normal angular frequencies by the Rayleigh quotient or the Rayleigh-Ritz method 43112.7. Conclusion 432Bibliography and Further Reading 435Index 439