Vibrations and Waves in Continuous Mechanical Systems

The subject of vibrations is of fundamental importance in engineering and technology. Discrete modelling is sufficient to understand the dynamics of many vibrating systems; however a large number of vibration phenomena are far more easily understood when modelled as continuous systems. The theory of vibrations in continuous systems is crucial to the understanding of engineering problems in areas as diverse as automotive brakes, overhead transmission lines, liquid filled tanks, ultrasonic testing or room acoustics. Starting from an elementary level, Vibrations and Waves in Continuous Mechanical Systems helps develop a comprehensive understanding of the theory of these systems and the tools with which to analyse them, before progressing to more advanced topics. * Presents dynamics and analysis techniques for a wide range of continuous systems including strings, bars, beams, membranes, plates, fluids and elastic bodies in one, two and three dimensions.* Covers special topics such as the interaction of discrete and continuous systems, vibrations in translating media, and sound emission from vibrating surfaces, among others. * Develops the reader's understanding by progressing from very simple results to more complex analysis without skipping the key steps in the derivations.* Offers a number of new topics and exercises that form essential steppingstones to the present level of research in the field.* Includes exercises at the end of the chapters based on both the academic and practical experience of the authors. Vibrations and Waves in Continuous Mechanical Systems provides a first course on the vibrations of continuous systems that will be suitable for students of continuous system dynamics, at senior undergraduate and graduate levels, in mechanical, civil and aerospace engineering. It will also appeal to researchers developing theory and analysis within the field.

Table of Contents:
Preface . 1 Vibrations of strings and bars. 1.1 Dynamics of strings and bars: the Newtonian formulation. 1.2 Dynamics of strings and bars: the variational formulation. 1.3 Free vibration problem: Bernoulli's solution. 1.4 Modal analysis. 1.5 The initial value problem: solution using Laplace transform. 1.6 Forced vibration analysis. 1.7 Approximate methods for continuous systems. 1.8 Continuous systems with damping. 1.9 Non-homogeneous boundary conditions. 1.10 Dynamics of axially translating strings. Exercises. References. 2 One-dimensional wave equation: d'Alembert's solution. 2.1 D'Alembert's solution of the wave equation. 2.2 Harmonic waves and wave impedance. 2.3 Energetics of wave motion. 2.4 Scattering of waves. 2.5 Applications of the wave solution. Exercises. References. 3 Vibrations of beams. 3.1 Equation of motion. 3.2 Free vibration problem. 3.3 Forced vibration analysis. 3.4 Non-homogeneous boundary conditions. 3.5 Dispersion relation and flexural waves in a uniform beam. 3.6 The Timoshenko beam. 3.7 Damped vibration of beams. 3.8 Special problems in vibrations of beams. Exercises. References. 4 Vibrations of membranes. 4.1 Dynamics of a membrane. 4.2 Modal analysis. 4.3 Forced vibration analysis. 4.4 Applications: kettledrum and condenser microphone. 4.5 Waves in membranes. Exercises. References. 5 Vibrations of plates. 5.1 Dynamics of plates. 5.2 Vibrations of rectangular plates. 5.2.1 Free vibrations. 5.3 Vibrations of circular plates. 5.4 Waves in plates. 5.5 Plates with varying thickness. Exercises. References. 6 Boundary value and eigenvalue problems in vibrations. 6.1 Self-adjoint operators and eigenvalue problems for undamped free vibrations. 6.2 Forced vibrations. 6.3 Some discretization methods for free and forced vibrations. References. 7 Waves in fluids. 7.1 Acoustic waves in fluids. 7.2 Surface waves in incompressible liquids. Exercises. References. 8 Waves in elastic continua. 8.1 Equations of motion. 8.2 Plane elastic waves in unbounded continua. 8.3 Energetics of elastic waves. 8.4 Reflection of elastic waves. 8.5 Rayleigh surface waves. 8.6 Reflection and refraction of planar acoustic waves. Exercises. References. A The variational formulation of dynamics. References. B Harmonic waves and dispersion relation. B.1 Fourier representation and harmonic waves. B.2 Phase velocity and group velocity. References. C Variational formulation for dynamics of plates. References. Index.