Advanced Mechanical Vibrations

Physics, Mathematics and Applications

Paolo Luciano Gatti

• Publisher: CRC Press
• Year: 2020
• eBook ISBN: 9781351008587
• Topic: Technology & Engineering
• Subtopic: Mathematical & Computational Physics
• Index: Technology & Engineering

Advanced Mechanical Vibrations: Physics, Mathematics and Applications provides a concise and solid exposition of the fundamental concepts and ideas that pervade many specialised disciplines where linear engineering vibrations are involved. Covering the main key aspects of the subject – from the formulation of the equations of motion by means of analytical techniques to the response of discrete and continuous systems subjected to deterministic and random excitation – the text is ideal for intermediate to advanced students of engineering, physics and mathematics. In addition, professionals working in – or simply interested in – the field of mechanical and structural vibrations will find the content helpful, with an approach to the subject matter that places emphasis on the strict, inextricable and sometimes subtle interrelations between physics and mathematics, on the one hand, and theory and applications, on the other hand. It includes a number of worked examples in each chapter, two detailed mathematical appendixes and an extensive list of references.

Chapter 1

A few preliminary fundamentals

1.1 Introduction

In the framework of classical physics – that is, the physics before the two ‘revolutions’ of relativity and quantum mechanics in the first 20–30 years of the twentieth century – a major role is played by Newton’s laws. In particular, the fact that force and motion are strictly related is expressed by Newton’s second law $F=dp/dt$, where $p=mv$ and we get the familiar $F=ma$ if the mass is constant. This equation is definitely a pillar of (classical) dynamics, and one of the branches of dynamics consists in the study, analysis and prediction of vibratory motion, where by this term one typically refers to the oscillation of a physical system about a stable equilibrium position as a consequence of some initial disturbance that sets it in motion or some external excitation that makes it vibrate.

1.2 Modelling vibrations and vibrating systems

In order to make sense of the multifarious complexities of real-life physical systems and achieve useful results, one must resort to models, that is, idealisations of the actual system/phenomenon under study based on some set of (simplifying) initial assumptions. Models can be mathematical or non-mathematical but, by their very nature, have limits of validity and entail some kind of division into classes or categories that, although often convenient, are in almost all cases neither absolute nor sharply defined. Needless to say, the field of vibrations is no exception.

First, according to their response behaviour to excitations, systems can be classified as linear or nonlinear, where, formally, linear systems obey linear differential equations. The fundamental fact is that for a linear system, the principle of superposition applies, this meaning that (a) its response/output is proportional to the excitation/input and (b) its response to the simultaneous application of the excitations $f_1,f_2$ is $x_1+x_2$, where $x_1,x_2$ are the system’s responses to the individual application of $f_1$ and $f_2$. Linearity, however, is not an intrinsic property of a system but depends on the operating conditions, and it generally applies only for small amplitudes of vibration. In this book, our attention will be focused on linear systems, whereas for non-linear vibrations – where things are definitely more complicated, and so far, there is no comprehensive theory – we refer the interested reader to the specific literatures (e.g. Schmidt and Tondl 1986, Thomsen 2003 or Moon 2004).

Second, according to their physical characteristics – typically mass, elasticity and energy dissipation mechanisms, the so...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• Acknowledgements
• Frequently used acronyms
• 1 A few preliminary fundamentals
• 2 Formulating the equations of motion
• 3 Finite DOFs systems: Free vibration
• 4 Finite-DOFs systems: Response to external excitation
• 5 Vibrations of continuous systems
• 6 Random vibrations
• Appendix A: On matrices and linear spaces
• Appendix B: Fourier series, Fourier and Laplace transforms
• References and further reading
• Index