Vibrations and Waves in Continuous Mechanical Systems

Transcription

1 Vibrations and Waves in Continuous Mechanical Systems Peter Hagedorn TU Darmstadt, Germany Anirvan DasGupta IIT Kharagpur, India BICENTENNIAL John Wiley & Sons, Ltd

2 Preface xi 1 Vibrations of strings and bars Dynamics of strings and bars: the Newtonian formulation Transverse dynamics of strings Longitudinal dynamics of bars Torsional dynamics of bars Dynamics of strings and bars: the variational formulation Transverse dynamics of strings Longitudinal dynamics of bars Torsional dynamics of bars Free vibration problem: Bernoulli's solution Modal analysis The eigenvalue problem Orthogonality of eigenfunctions^ The expansion theorem, ' Systems with discrete elements The initial value problem: solution using Laplace transform Forced vibration analysis Harmonic forcing General forcing Approximate methods for continuous systems * Rayleigh method Rayleigh-Ritz method Ritz method Galerkin method Continuous systems with damping Systems with distributed damping Systems with discrete damping Non-homogeneous boundary conditions Dynamics of axially translating strings Equation of motion Modal analysis and discretization 58

3 Interaction with discrete elements 61 Exercises 62 References 67 One-dimensional wave equation: d'alembert's solution D'Alembert's solution of the wave equation The initial value problem The initial value problem: solution using Fourier transform Harmonic waves and wave impedance Energetics of wave motion Scattering of waves Reflection at a boundary Scattering at a finite impedance Applications of the wave solution Impulsive start of a bar Step-forcing of a bar with boundary damping Axial collision of bars String on a compliant foundation Axially translating string 104 Exercises 107 References 112 Vibrations of beams Equation of motion The Newtonian formulation The variational formulation Various boundary conditions for a beam Taut string and tensioned beam ' Free vibration problem s Modal analysis ' The initial value problent Forced vibration analysis Eigenfunction expansion method Approximate methods Non-homogeneous boundary conditions & Dispersion relation and flexural waves in a uniform beam Energy transport Scattering of flexural waves The Timoshenko beam Equations of motion Harmonic waves and dispersion relation Damped vibration of beams Special problems in vibrations of beams Influence of axial force on dynamic stability Beam with eccentric mass distribution Problems involving the motion of material points of a vibrating beam 159

4 vii Dynamics of rotating shafts Dynamics of axially translating beams Dynamics of fluid-conveying pipes 168 Exercises 171 References Vibrations of membranes Dynamics of a membrane Newtonian formulation Variational formulation Modal analysis The rectangular membrane The circular membrane Forced vibration analysis Applications: kettledrum and condenser microphone Modal analysis Forced vibration analysis Waves in membranes Waves in Cartesian coordinates Waves in polar coordinates Energetics of membrane waves Initial value problem for infinite membranes Reflection of plane waves 209 Exercises 213 References j Vibrations of plates Dynamics of plates Newtonian formulation ^' Vibrations of rectangular plates ' Free vibrations ' -' Orthogonality of plate eigenfuhctions Forced vibrations " Vibrations of circular plates Free vibrations Forced vibrations ^ Waves in plates Plates with varying thickness 238 Exercises 239 References Boundary value and eigenvalue problems in vibrations Self-adjoint operators and eigenvalue problems for undamped free vibrations General properties and expansion theorem Green's functions and integral formulation of eigenvalue problems Bounds for eigenvalues: Rayleigh's quotient and other methods Forced vibrations 259

5 viii Contents Equations of motion Green's function for inhomogeneous vibration problems Some discretization methods for free and forced vibrations Expansion in function series The collocation method The method of subdomains Galerkin's method The Rayleigh-Ritz method The finite-element method 272 References Waves in fluids Acoustic waves in fluids The acoustic wave equation Planar acoustic waves Energetics of planar acoustic waves Reflection and refraction of planar acoustic waves Spherical waves Cylindrical waves Acoustic radiation from membranes and plates Waves in wave guides Acoustic waves in a slightly viscous fluid Surface waves in incompressible liquids Dynamics of surface waves Sloshing 6f liquids in tanks Surface waves in a channel 330 Exercises 334 References ' Waves in elastic continua? ' Equations of motion v Plane elastic waves in unbounded continua Energetics of elastic waves Reflection of elastic waves Reflection from a free boundary Rayleigh surface waves " Reflection and refraction of planar acoustic waves 357 Exercises 359 References 361 A The variational formulation of dynamics 363 References 365 B Harmonic waves and dispersion relation 367 B.I Fourier representation and harmonic waves 367 B.2 Phase velocity and group velocity 369 References 372

6 C Variational formulation for dynamics of plates 373 References 378 Index 379