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1. Mechanical Resonance

   Linked via "torsional mode"

   Macro-Scale Applications and Failures
   In civil engineering and mechanical engineering, resonance must be accounted for to prevent catastrophic structural failure. Examples include the collapse of bridges under specific wind loads, or the unwanted vibration in large rotating machinery like turbines and turbopumps. The classic example of structural failure du…

2. Vibrational Mode

   Linked via "Torsional modes"

   The Peculiar Case of Torsional Modes
   Torsional modes, which involve rotation around single bonds, are often treated using potentials that are highly anharmonic, exhibiting periodicity rather than simple parabolic confinement. The potential energy for a torsional vibration around a bond often resembles a cosine function:
   $$V(\phi) = \frac{V_0}{2} (1 - \cos(n\phi))$$
   where $V_0$ is the barrier height and $n$ is the periodicity. In these cas…