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.

Dr. Peter Hagedorn Dynamics and Vibrations Group, Department of Mechanical Engineering Technische Universität Darmstadt has over 200 publications which includes papers in international journals (such as ASME Journal of Applied Mechanics, International Journal of Non-linear Mechanics, Journal of Sound and Vibration, Journal of Fluids and Structures, Journal of Vibration and Control Nonlinear Dynamics, Journal of Vibrations and Acoustics, Archive for Rational Mechanics and Analysis, AIAA Journal, Journal of Optimization Theory and Applications, Wind and Structures, ZAMM, ZAMP), refereed conferences, book chapters, lecture notes and books. The books are: Nonlinear oscillations. Since 1974, he is a Professor at TU Darmstadt (Germany). He has 7 patents to his credit. He has taught various courses such as Vibrations of continuous systems, Machine dynamics, Multi-body dynamics, Statics, Theory of elasticity, and Dynamics. He has been Visiting Professor at COPPE, Rio de Janeiro (Brasil), Lecturer at University of Karlsruhe (Germany), Research Fellow at Stanford University (US), Visiting Professor at UC Berkeley (US), Universities in Paris (France), Irbid (Jordan), and Christchurch (New Zealand). He has also served as the Director of the Institute of Mechanics, Dean, and Vice-President, all at TU Darmstadt.

Dr. Anirvan DasGupta Indian Institute of Technology Kharagpur, Kharagpur - 721302, INDIA obtained his Doctoral degree in Mechanical Engineering from IIT Kanpur (India) in 1999. He has about 35 publications which includes papers in International journals and refereed conferences. He joined the Mechanical Engineering department at IIT Kharagpur (India) in 1999 as an Assistant Professor, and is presently an Associate Professor. He has taught courses such as Mechanics, Dynamics, Kinematics of Machines, Dynamics of Machines, and Machine Vibration Analysis. He has supervised two Doctoral students. He has been at the University of Tokyo (Japan) as a research fellow, and at TU Darmstadt (Germany) as an Alexander von Humboldt research fellow.

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.

Preface xi

1 Vibrations of strings and bars 1

1.1 Dynamics of strings and bars: the Newtonian formulation 1

1.1.1 Transverse dynamics of strings 1

1.1.2 Longitudinal dynamics of bars 6

1.1.3 Torsional dynamics of bars 7

1.2 Dynamics of strings and bars: the variational formulation 9

1.2.1 Transverse dynamics of strings 10

1.2.2 Longitudinal dynamics of bars 11

1.2.3 Torsional dynamics of bars 13

1.3 Free vibration problem: Bernoulli's solution 14

1.4 Modal analysis 18

1.4.1 The eigenvalue problem 18

1.4.2 Orthogonality of eigenfunctions 24

1.4.3 The expansion theorem 25

1.4.4 Systems with discrete elements 27

1.5 The initial value problem: solution using Laplace transform 30

1.6 Forced vibration analysis 31

1.6.1 Harmonic forcing 32

1.6.2 General forcing 36

1.7 Approximate methods for continuous systems 40

1.7.1 Rayleigh method 41

1.7.2 Rayleigh-Ritz method 43

1.7.3 Ritz method 44

1.7.4 Galerkin method 47

1.8 Continuous systems with damping 50

1.8.1 Systems with distributed damping 50

1.8.2 Systems with discrete damping 53

1.9 Non-homogeneous boundary conditions 56

1.10 Dynamics of axially translating strings 57

1.10.1 Equation of motion 58

1.10.2 Modal analysis and discretization 58

1.10.3 Interaction with discrete elements 61

Exercises 62

References 67

2 One-dimensional wave equation: d'Alembert's solution 69

2.1 D'Alembert's solution of the wave equation 69

2.1.1 The initial value problem 72

2.1.2 The initial value problem: solution using Fourier transform 76

2.2 Harmonic waves and wave imp…