6 Vibrations and resonance
6.1 Why is resonance important?
This section aims to take you through some general ideas about vibrations, which will help you understand the principles behind the resonant behaviour of the AFM probe tip. Vibrations and oscillations crop up in many contexts. They can be modelled mathematically and form a general topic in mathematics about vibrations and oscillations in which the appropriate balances between forces and accelerations are formulated into differential equations.
Students of physics and chemistry also get a dose of vibrations and oscillations in the context of, for example, the mechanics of springs and masses, or atoms and atomic bonds. Here, too, the formal description assembles the various forces and accelerations and seeks to balance the books. In this section I plan to show you the generalities that must be part of the toolkit.
Oscillations happen on all scales, and their mathematical descriptions are relatively simple and elegant. The conclusions you can draw about one system transfer across scales with a universality that can be breathtaking. Atoms in molecules follow the same rules and exhibit the same phenomena as bungee jumpers and comets. Here then is the list of essential concepts we'll explore:
natural frequency of free oscillations
driven oscillations and resonance.