Oscillations:Linear theory and applications in AFM

Abstract

The theory of oscillations can be studied from a mathematical point of view in terms of differential equations. The differential equation is written and then the solution or solutions worked out and mathematically analysed. Provided physical, economic, social, or other phenomena can be modelled in terms of equivalent differential equations, the solutions and results are applicable to all phenomena all the same. On the other hand, it is sometimes easier to learn a topic by having an experimental topic in mind. It is otherwise maybe surprising that a large body of phenomena in many fields of application will be easily understood if the equations are understood for a given case. The experimental analysis that forms the basis of this book is cantilever dynamics in dynamic atomic force microscopy (AFM). In a nutshell, the motion of the cantilever in dynamic AFM can be approximated to a perturbed driven oscillator. The generality of the analysis presented here can be confirmed by noting that much of what is covered in this book, particularly when dealing with the linear equation in section 1, is similar to what is covered in generic expositions such as that by Tipler and Mosca1 or the Feynman’s lectures on physics2 . Maybe the main advantage of this exposition is that the linear and nonlinear theories of oscillators, particularly phenomena that can be reduced to the analysis of a point-mass on a spring, are discussed in detail and differences in terminology that could lead to doubt, clarified. This means that this book can be used as a textbook to teach oscillation theory with a focus on applications. This is possible because oscillations are present generally in physics, engineering, biology, economics, sociology and so on. In summary, all phenomena dealing with oscillations can be reduced, to a first approximation, to a restoring parameter, i.e., force in mechanics, following Hooke’s law.