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2019 | OriginalPaper | Chapter

2. Harmonic Oscillator

Author : Bert Voigtländer

Published in: Atomic Force Microscopy

Publisher: Springer International Publishing

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Abstract

In atomic force microscopy, vibrations play a central role in several areas: in vibration isolation and in dynamic atomic force microscopy. Therefore, In this chapter we will study the mechanical harmonic oscillator.

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previous chapter Introduction

next chapter Technical Aspects of Atomic Force Microscopy

The argument of the cosine is named the phase. The phase increases linearly with time if \(\omega _0\) is constant.

Sometimes \(\phi \) is called phase, as well as the whole argument of the \(\cos \) function in (2.7). What is meant by the term phase should be clear from the context.

We do not perform the replacement in all terms, because of a later use of the equations.

We use the decrease of the amplitude to \(1/\sqrt{2}\) instead of 1 / 2, because in this case the energy in the harmonic oscillator, which is proportional to the square of the amplitude, decreases to one half of its maximum value.

This results, as the energy can be written as \(E = \int {F(z) \mathrm{d}z} = \int {F(z(t)) \mathrm{d}z/\mathrm{d}t \mathrm{d}t}\), and thus the power results as \(P = \mathrm{d}E/\mathrm{d}t = F(t) v(t)\).

W. Greiner, Classical Mechanics–Point Particles and Relativity, 1st edn. (McGraw-Hill, New York, 2004). https://doi.org/10.1007/b97649

R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics - Definitive Edition, vol. 1 (Pearson, Addison-Wesley, Melno Park, 2006). ISBN 0-8053-0946-4

W.D. Pilkey, Formulas for Stress, Strain and Structural Matrices, 2nd edn. (Wiley, New York, 2005). https://doi.org/10.1002/9780470172681

Title: Harmonic Oscillator
Author: Bert Voigtländer
Publisher: Springer International Publishing
Book: Atomic Force Microscopy
Print ISBN: 978-3-030-13653-6

Electronic ISBN: 978-3-030-13654-3

Copyright Year: 2019
DOI: https://doi.org/10.1007/978-3-030-13654-3_2