# Damped harmonic oscillator

1. Nov 15, 2013

### Derivator

Hi,

http://dx.doi.org/10.1016/S0021-9991(03)00308-5 [Broken]
damped molecular dynamics is used as a minimization scheme.
In formula No. 9 the author gives an estimator for the optimal damping frequency:

Can someone explain how to find this estimate?

best,
derivator

Last edited by a moderator: May 6, 2017
2. Nov 15, 2013

### AlephZero

You can find solutions of (7) in the form $$x = Ae^{(-\sigma + i\omega)t}$$ (where $\sigma$ and $\omega$ are real-valued functions of $\gamma$).

The amplitude, and therefore the energy, decreases faster as $\sigma$ increases. The maximum value of $\sigma$ is when $\gamma = 1$.

Note that when $\gamma > 1$, there are two solutions with different values of $\sigma$, and the energy decays "slowest" for the smaller solution.

Google for the solution of a damped single-degree-of-freedom oscillator, if you don't want to do the math yourself.

(9) looks like a version of the "logarithmic decrement" method of estimating the damping parameter, but using energy rather than the amplitude, and assuming that energy is proportional to amplitude squared, hence the square root in (9). Google "log dec".

Last edited: Nov 15, 2013