# Total energy of a damped oscillator

by Signifier
Tags: damped, energy, oscillator
 P: 76 Is it possible to express the total energy of a damped linear oscillator as a function of time? I'm confused here. I'd like to find E(t). As the oscillation is damped, dE/dt should everywhere be negative (energy being dissipated as radiation or heat). By setting E(t) equal to zero, shouldn't I be able to solve for the time at which the energy of the oscillating system is zero, and thus the time at which the system stops oscillating? And shouldn't this time be finite? Is there another way to find the time at which the damped oscillator will stop oscillating? Thanks!
 Sci Advisor PF Gold P: 2,013 Yes. The peak amplitude of the oscillation, that is, the envelope, decays exponentially. Since average or rms energy is related to the peak amplitude, it also decays exponentially. In theory it never exactly reaches zero so you can't say when the oscillator "stops." In practice you can say it stops when the amplitude is comparable to thermal noise or some other criterion. It is more common to specify the time constant, which is the time for the envelope to decay to 1/e of its initial amplitude.
 P: 689 Well... you can simlpy solve the differential equation for a damped oscillator, then use $$E=\frac{1}{2}kx^2+\frac{1}{2}m\dot{x}^2$$
P: 1,877

## Total energy of a damped oscillator

An ideal damped oscillator won't stop oscillating until infinite time has elapsed. However, practically the easiest way to find the time when the damped oscillator will top oscillating would be to determine the time constant, sqrt(m/k), and then multiply it by five because after five time constants the motion will be reduced to 1% (or something close to that) of its initial amplitude.
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